Multiple solution sets

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. 

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)... 

One lesson from this example is that one must typically have sufficient insight into a physical problem to recognize when a physically valid solution has been obtained or not obtained. Many nonlinear equation sets have multiple solution sets and only one set may have meaning for a particular physical problem. 

Hence there are multiple solutions for the final set of 10000 compounds. The final selection can be diversity driven using for example cluster analysis based on multiple fingerprints [63], hole filling strategies by using scaffold/ring analysis (LeadScope [66], SARVision [66]) or pharmacophore analysis [67, 68]. For a review of computational approaches to diversity and similarity-based selections, see the paper of Mason and Hermsmeier [69] and the references therein. 

Nf = 0 The problem is exactly determined. If NF = 0, then the number of independent equations is equal to the number of process variables and the set of equations may have a unique solution, in which case the problem is not an optimization problem. For a set of linear independent equations, a unique solution exists. If the equations are nonlinear, there may be no real solution or there may be multiple solutions. 

The energy and spectral optimization problems are convex programs so when there are multiple solutions the solution sets form a convex set. The following corollary characterizes how these convex sets of solutions relate to solutions of the Euler equation. In the formulation of this corollary we use the notion of optimal gap Ao—the gap achieved by optimal P and S. The optimal gap is a characteristic of the energy problem, depending only on H and S. 

Equations (260) (263) form a set of differential-algebraic equations which has a unique solution when the two algebraic equations [(261) and (263)] themselves have unique solution of (c) (and Or)) for any fixed cm (and 0lm). Equivalently, the above system has multiple solutions only when Eqs. (261) and (263) evaluated at the reactor exit conditions begin to have multiple solutions. For Le - 1 (typical fluid Lewis numbers vary between 1 and 100), and for y oo, the hysteresis variety for the above set of equations is given by... 

It should be noted that for non-isothermal case (and also for isothermal case with autocatalytic kinetics) the local equation may have multiple solutions. When this occurs, the averaged model obtained by the L-S method captures the complete set of solutions of the full CDR equations only within the region of convergence of the local equation. For example, for the wall-catalyzed non-isothermal reaction case, we have shown that the averaged two-mode model can capture only the three azimuthally symmetric solutions of the full CDR equation. The latter has three symmetric solutions (of which two are stable) as... 

Two models are available for interpreting attenuation spectra as a PSD in suspensions with chemically distinct, dispersed phases using the extended coupled phase theory.68 Both models assume that the attenuation spectrum of a mixture is composed of a superposition of component spectra. In the multiphase model, the PSD is represented as the sum of two log-normal distributions with the same standard deviation, that is, a bimodal distribution. The appearance of multiple solutions is avoided by setting a common standard deviation to the mean size of each distribution. This may be a poor assumption for the PSD (see section 11.3.2). The effective medium model assumes that only one target phase of a multidisperse system needs to be determined, while all other phases contribute to a homogeneous system, the so-called effective medium. Although not complicated by the possibility of multiple solutions, this model requires additional measurements to determine the density, viscosity, and acoustic attenuation of the effective medium. The attenuation spectrum of the effective medium is modeled via a polynomial fit, while the target phase is assumed to have a log-normal PSD.68 This model allows the PSD for mixtures of more than two phases to be determined. 

Suppose we have a mixture of water, pyridine, and toluene. We set the pressure to 1 atm and attempt to solve the above equations. Lacking any further insight, we set all the vapor and liquid compositions equal to 0.33333 and then attempt a solution using a Newton-based method. The problem with finding azeotropes becomes immediately evident. There are six solutions to these equations the three binary azeotropes and the three pure species. There is no ternary azeotrope. To find all azeotropes for a mixture, we must find all solutions to the above equations. Finding multiple solutions to a set of highly nonlinear equations like these is usually a very difficult task. 

Another fact to keep in mind is that the observed data is also a single random realization of a probability distribution. This means that the model may provide a good description of this distribution but that there is an apparent misfit only because of sample variability. Again, the solution is to simulate multiple data sets from the model and use the multiply predicted probabilities to construct prediction intervals. If the model provides an appropriate description of the observed data, we would expect the lines connecting the cumulative probabilities to be included in... 

In three-dimensional systems there can be even more interesting examples of multiple solutions. In addition, some of these systems (models) may exhibit different sets of multiple solutions for different intervals of parameters. The most complete analysis of such a case is given in Gurel and Rossler (1979). 

The solution set of values for the v s is readily obtained by application of the multiplication rule beginning with the first row. Originally, the component-material balances for the separated lights were formulated by enclosing the condenser-accumulator section and each plate in succession, r, r + 1, r + 2, s — 3, s — 2, and s — l.4 The equations so obtained are equivalent to the set given by Eq. (11-12) see Prob. 11-7. 

Use simple criteria such as the one-quarter decay ratio (see Example 16.1), minimum settling time, minimum largest error, and so on. Such an approach is simple and easily implementable on an actual process. Usually, it provides multiple solutions (see Example 16.1). Additional specifications on the closed-loop performance will then be needed to break the multiplicity and select a single set of values for the adjusted parameters. 

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