Box-Wilson optimization method

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... 

In this study, ethanol extraction using SC CO2 was achieved from 15 to 75 %(v/v) synthetic ethanol solutions and also from fermentation broth. Effects of temperature, pressure, extraction time and initial ethanol concentration on extraction yield were investigated in the ranges of 313 to 333 K and 80 to 160 atmospheres. Optimum conditions for the batch extraction of 15%(v/v) ethanol solution were found using the Box-Wilson optimization method in a linear form. In addition, the effect of consecutive solvent feeding on extraction yield were investigated both with 15 %(v/v) ethanol solution and with fermentation broth. 

Initial screens can be distinguished between methods that are used to determine what factors are most important, and follow-up screens that allow optimization and improvement of crystal quality (Table 14.1). In experimental design, this is known as the Box-Wilson strategy (Box et al., 1978). The first group of screens is generally based on a so-called factorial plan which determines the polynomial coefficients of a function with k variables (factors) fitted to the response surface. It can be shown that the number of necessary experiments n increases with 2 if all interactions are taken into account. Instead of running an unrealistic, large number of initial experiments, the full factorial matrix can... 

Statistical optimization methods other than the Simplex algorithm have only occasionally been used in chromatography. Rafel [513] compared the Simplex method with an extended Hooke-Jeeves direct search method [514] and the Box-Wilson steepest ascent path [515] after an initial 23 full factorial design for the parameters methanol-water composition, temperature and flowrate in RPLC. Although they concluded that the Hooke-Jeeves method was superior for this particular case, the comparison is neither representative, nor conclusive. 

In literature, inconsistent results were given with respect to the effect of temperature and pressure on extraction yield at low ethanol concentrations [4,7,12]. For this reason, an experimental design was performed utilizing the Box-Wilson method to observe this effect more clearly and for the optimization of the experimental conditions. 

This section introduces the two most common empirical optimization strategies, the simplex method and the Box-Wilson strategy. The emphasis is on the latter, as it has a wider scope of applications. This section presents the basic idea the techniques needed at different steps in following the given strategy are given in the subsequent sections. 

The technique allows immediate interpretation of the regression equation by including the linear and interaction (cross-product) terms in the constant term (To or stationary point), thus simplifying the subsequent evaluation of the canonical form of the regression equation. The first report of canonical analysis in the statistical literature was by Box and Wilson [37] for determining optimal conditions in chemical reactions. Canonical analysis, or canonical reduction, was described as an efficient method to explore an empirical response surface to suggest areas for further experimentation. In canonical analysis or canonical reduction, second-order regression equations... 

By far the most popular technique is based on simplex methods. Since its development around 1940 by DANTZIG [1951] the simplex method has been widely used and continually modified. BOX and WILSON [1951] introduced the method in experimental optimization. Currently the modified simplex method by NELDER and MEAD [1965], based on the simplex method of SPENDLEY et al. [1962], is recognized as a standard technique. In analytical chemistry other modifications are known, e.g. the super modified simplex [ROUTH et al., 1977], the controlled weighted centroid , the orthogonal jump weighted centroid [RYAN et al., 1980], and the modified super modified simplex [VAN DERWIEL et al., 1983]. CAVE [1986] dealt with boundary conditions which may, in practice, limit optimization procedures. 

The method of steepest ascent determines the direction from an initial experimental domain which has the steepest slope upwards along the response surface, and hence point towards the optimum conditions. A series of experiments can then be run along this steepest ascent vector. This will lead to rapid improvements. The method was introduced by Box and Wilson[l] and was the first method for systematic multivariate optimization experiments in chemistry. 

Response surface methodology (RSM) is a method of optimization using statistical techniques based upon the special factorial designs of Box and Behenkini and Box and Wilson.I It is a scientific approach to determining optimum conditions which combines special experimental designs with Taylor first and second order equations. The RSM process determines the surface of the Taylor expansion curve which describes the response (yield, impurity level, etc.) The Taylor equation, which is the heart of the RSM method, has the form ... 

---