Optimum region

From Table V it can be seen that the last vertices in the simplex are not always the ones with the best response. This is due to the fact that once an optimum region is reached, the simplex may begin to step outside of the optimum and circle" it, sporadically moving back to the optimum or a near-optimum every few steps. This is precisely what is illustrated in Table V, where the boxed-in vertices represent the six best results. Statistical treatment of these data indicate that a consensus has more or less been reached for the optimum conditions, particularly the initial density and initial temperature. One should not infer, however, that the mean values for all 4 parameters should be used instead of the (best) values corresponding to vertex 13. We have yet to perform sufficient comparisons to draw such a conclusion. 

When selecting a domain of factors one should pay special attention to choosing the center point of the design (basic level or the null point). The choice of a null point is associated with selection of the initial status of the research subject to perform optimization. As optimization is connected with improvement of the subject status in comparison with the status in the null point, it is desirable that the point is in the optimum region or as close to it as possible. If the mentioned research was preceded by other experiments on the same subject, the status having the most convenient response value is taken as the null experiment. The null point is quite often the center of the domain of factors. The most important alternatives in selecting the basic and null levels are shown in Fig. 2.10. 

In this way the basic experiment is defined for the linear model, and the gradient that indicates the direction of the fastest response increase or decrease is obtained. When a response maximum or minimum is searched for, the experimental center is moved that way and a new experiment for the linear model performed. The procedure is repeated until moving along the gradient has an effect. When this has no effect, it means we are close to the optimum. Polynomials of higher order, mostly the second, are used in the optimum region. 

When the optimum region is close by, three solutions are possible ... 

An increase in the number of replicated trials causes a decrease in reproducibility variance or experimental error as well as in the associated variances of regression coefficients. Design points-trials can be replicated in all points of the experiment or in some of them. An upgrade of the design of experiment may be realized by a shift from fractional to full factorial experiment, a switch to bigger replica (from 1/6 to 1 /2 replica), a switch to second-order design (when the optimum region is dose by), etc. 

All regression coefficients are statistically significant and the optimum region is close by, so that the best obtained response value is y=95%. The research objective is to obtain 99 to 100% of neodymium with a minimal number of trials. 

If the optimum region is close by, the research by this model ends and we switch to constructing the design of experiments for the second-order model. Fig. 2.38 shows the block diagram of searching for an optimum for an inadequate linear model. 

In Example 2.26, we have obtained the linear regression model for dynamic viscosity y, P, as a function of mixing speed X3, min"1 and mixing time X2, min of composite rocket propellant. To determine the conditions of minimal viscosity, a method of steepest ascent has been applied. This method has defined the local optimum region that has to be described by a second-order model. Conditions of the factor variations are shown in Table 2.146. 

With nonlinear models, which are aimed at mathematical modeling or adequate description of the optimum region and that as a rule have numerous regression coefficients, rejection of insignificant regression coefficients is not so important as in the phase of linear modeling. For second-order models, an estimate of lack of fit or inadequacy of the model is of particular importance. 

At that location it may be necessary to fix the positions of another design and to acquire missing data to find a higher order polynomial and to build the final model of the near-optimum region. 

An enzymatic process cannot be adopted for industrial production of BDF, if immobilized lipase cannot be used for long period. Studies on the stability of immobilized C. antarctica lipase revealed that it was the most stable when the reaction was conducted with 5 to 8 mol MeOH for total FAs in acid oil prepared from soapstock (Watanabe et al., 2007). The phenomenon, that there is the optimum region of MeOH concentration, may be explained as follows ... 

Site density and pellet size in FT synthesis catalysts may not be entirely within the control of the catalyst designer. Low site densities may be required in order to bring catalysts limited by CO transport into the optimum region of x values, a modification that unfortunately also lowers reactor productivity. Pellet size may be decreased instead in order to eliminate CO... 

Fig. 9 Superposition of contour plots for turbidity <3 ppm and cloud point <70° to determine an optimum region ( slice at 4.3% polysorbate 80). Compare with Fig. 5. (From Ref l)...

The approach of using a mathematical model to map responses predictively and then to use these models to optimize is limited to cases in which the relatively simple, normally quadratic model describes the phenomenon in the optimum region with sufficient accuracy. When this is not the case, one possibility is to reduce the size of the domain. Another is to use a more complex model or a non-polynomial model better suited to the phenomenon in question. The D-optimal designs and exchange algorithms are useful here as in all cases of change of experimental zone or mathematical model. In any case, response surface methodology in optimization is only applicable to continuous functions. 

The first two methods are useful for locating a near-optimum region, but they are not well suited for a more precise location of the specific optimum. 

Response surface modelling is used to locate the detailed optimum conditions. The principle is to establish a response surface model which maps the optimum region. The map can then be used for navigation in the optimum region. Close to an optimum the surface is curved and to describe the general features of the response surface quadratic models are used. The response surface methods have been and are being developed by Box and coworkers.[3]... 

There is, however, one application when the overall desirability D can be rather safely used, namely as a tool in conjunction with response surface modelling. In this context, it can be used to explore the joint modelling of several responses so that a near-optimum region can be located by simulations against the response surface models. The search for conditions which incrase D can be effected either by simplex techniques or by the method of steepest ascent. For the steepest ascent, a linear model for D is first determined from the experiments in the design used to establish the response surface models. The settings which increase D can be translated back into the individual responses by using the response surface models. Thus, it is possible to establish immediately whether the simulated reponse values correspond to suitable experimental conditions. Such results must, of course, be verified by experimental runs. 

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