Optimization linear models

The Box-Cox parameter is set at -I = 1 for using y as dependent variable or A-0 for using Iny as dependent variable. A search for the independent variables to be included in the optimal linear model is carried out using SROV. The search is repeated with o set at zero and the best model is selected according to the variance and R values. The residual plot for the selected model is examined. If the errors are randomly distributed finish, otherwise proceed to step 2. 

Confidence limits for the parameter estimates define the region where values of bj are not significantly different from the optimal value at a certain probability level 1-a with all other parameters kept at their optimal values estimated. The confidence limits are a measure of uncertainty in the optimal estimates the broader the confidence limits the more uncertain are the estimates. These intervals for linear models are given by... 

For linear models the joint confidence region is an Alp-dimensional ellipsoid. All parameters encapsulated within this hyperellipsoid do not differ significantly from the optimal estimates at the probability level of 1-a. 

The nature of the optimization problem can mm out to be linear or nonlinear depending on the mass transfer model chosen14. If a model based on a fixed outlet concentration is chosen, the model turns out to be a linear model (assuming linear cost models are adopted). If the outlet concentration is allowed to vary, as in Figure 26.35a and Figure 26.35b, then the optimization turns out to be a nonlinear optimization with all the problems of local optima associated with such problems. The optimization is in fact not so difficult in practice as regards the nonlinearity, because it is possible to provide a good initialization to the nonlinear model. If the outlet concentrations from each operation are initially assumed to go to their maximum outlet concentrations, then this can then be solved by a linear optimization. This usually... 

The well-known Box-Wilson optimization method (Box and Wilson [1951] Box [1954, 1957] Box and Draper [1969]) is based on a linear model (Fig. 5.6). For a selected start hyperplane, in the given case an area A0(xi,x2), described by a polynomial of first order, with the starting point yb, the gradient grad[y0] is estimated. Then one moves to the next area in direction of the steepest ascent (the gradient) by a step width of h, in general... 

Nf > 0 The problem is underdetermined. If NF > 0, then more process variables exist in the problem than independent equations. The process model is said to be underdetermined, so at least one variable can be optimized. For linear models, the rank of the matrix formed by the coefficients indicates the number of independent equations (see Appendix A). 

The targets for the MPC calculations are generated by solving a steady-state optimization problem (LP or QP) based on a linear process model, which also finds the best path to achieve the new targets (Backx et al., 2000). These calculations may be performed as often as the MPC calculations. The targets and constraints for the LP or QP optimization can be generated from a nonlinear process model using a nonlinear optimization technique. If the optimum occurs at a vertex of constraints and the objective function is convex, successive updates of a linearized model will find the same optimum as the nonlinear model. These calculations tend to be performed less frequently (e.g., every 1-24 h) due to the complexity of the calculations and the process models. 

KINPTR is also used extensively for commercial planning guidance and reformer operation optimization. The model has been used to improve linear program accuracy for short-range refinery planning and crude oil supply and distribution studies. It is also used to optimize reformer economics by proper selection of operating conditions within refinery constraints. 

Similarly, the degree of freedom available to optimize transfer prices depends on factors outside the model such as accumulated deficits or the profitability of other business activities in a country. Furthermore, legal constructs such as principal trading companies (cf. Murphy 1998) are employed in practice to optimally allocate profits within a company. Finally, from a modeling perspective the simultaneous optimization of network design and transfer prices leads to a non-linear model (cf. Schmidt and Wilhelm 2000, p. 1510 Verter and Dincer 1995, p. 278) that is difficult to solve to optimality for realistic problem instances. Therefore, transfer prices are not optimized but instead determined outside the optimization model. Furthermore, transfer prices are assumed to be independent of the transfer destination because tax authorities in the country of origin usually do not accept differentiated transfer prices. In literature differentiated approaches can nevertheless be found as well (e.g., Kouvelis et al. 2004, p. 130). 

Movement to optimum by an inadequate linear model is also possible in cases when doing the mentioned eight trials is not acceptable. The values of linear regression coefficients are considerably above the values of those for interactions, the more so since linear effects are not aliased/confounded with interaction effects. Although the movement to optimum by an inadequate linear model is mathematically incorrect, it may be accepted in practice with an adequate risk. Note that when trying to optimize a process one should aspire towards both the smallest possible interaction effects and approximate or symmetrical linear coefficients. In problems of interpolation models, the situation is exactly the opposite since it insists on interaction effects, which may be significant. 

A process having properties dependent on four factors has been tested. A full factorial experiment and optimization by the method of steepest ascent have brought about the experiment in factor space where only two factors are significant and where an inadequate linear model has been obtained. To analyze the given factor space in detail, a central composite rotatable design has been set up, as shown in Table 2.152. 

The linear regression model is inadequate with 95% confidence. Since the linear model is neither symmetrical nor adequate and since the application of the method of steepest ascent would lead to a one-factor optimization (b2 is by far the greatest), a new FRFE 24 1 has been designed with doubled variation intervals for X3 X3 and X4. 

This chapter gave an overview of how to simplify complex processes sufficiently to allow the use of analytical models for their analysis and optimization. These models are based on mass, momentum, energy and kinetic balance equations, with simplified constitutive models. At one point, as the complexity and the depth of these models increases by introducing more realistic geometries and conditions, the problems will no longer have an analytical solution, and in many cases become non-linear. This requires the use of numerical techniques which will be covered in the third part of this book, and for the student of polymer processing, perhaps in a more advanced course. 

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