Simplex method optimisation

We will present the basics of the simplex method with the aid of a simulation and then describe the algorithm. As an example, Soylak et al. [18] optimised a procedure to preconcentrate lead (the studied response, Y) using a 2 factorial design in which the factors were ... 

J. Echeverria, M. T. Arcos, M. J. Fernandez and J. Garrido Segovia, Simplex method strategy for optimisation in flames atomic absorption spectrometry, Quim. Anal., 11(1), 1992, 25-33. 

G. H. Lee, Development of new high temperature plasma sources for spectrochemical analysis multivariate optimisation by the modified sequential simplex method. Bull. Korean Chem. Soc., 14(2), 1993, 275-281. 

G. A. Zachariadis and J. A. Stratis, Optimisation of cold vapour atomic absorption spectrometric determination of mercury with and without amalgamation by subsequent use of complete and fractional factorial designs with univariate and modified simplex methods, J. Anal. At. Speetrom., 6(3), 1991, 239-245. 

It is always preferable to optimise by varying all the factors simultaneously. The latter approach leads to experimental design and to the simplex method, which is described in a number of specialised monographs. 

Mayur et al. (1970) formulated a two level dynamic optimisation problem to obtain optimal amount and composition of the off-cut recycle for the quasi-steady state operation which would minimise the overall distillation time for the whole cycle. For a particular choice of the amount of off-cut and its composition (Rl, xRI) (Figure 8.1) they obtained a solution for the two distillation tasks which minimises the distillation time of the individual tasks by selecting an optimal reflux policy. The optimum reflux ratio policy is described by a function rft) during Task 1 when a mixed charge (BC, xBC) is separated into a distillate (Dl, x DI) and a residue (Bl, xBi), followed by a function r2(t) during Task 2, when the residue is separated into an off-cut (Rl, xR2) and a bottom product (B2, x B2)- Both r2(t)and r2(t) are chosen to minimise the time for the respective task. However, these conditions are not sufficient to completely define the operation, because Rl and xRI can take many feasible values. Therefore the authors used a sequential simplex method to obtain the optimal values of Rl and xR which minimise the overall distillation time. The authors showed for one example that the inclusion of a recycled off-cut reduced the batch time by 5% compared to the minimum time for a distillation without recycled off-cut. 

There is some controversy as to whedier simplex methods should genuinely be considered as experimental designs, rather than algorithms for optimisation. Some statisticians often totally ignore this approach and, indeed, many books and courses of... 

Sequential optimisation methods are used for multi-parameter optimisation. The simplex method starts with some initial experiments, evaluates from them the values of a sum optimisation criterion (COF), on the basis of these results determines the next combination of operation parameters to be used for running a new chromatographic experiment and compares the value of the COF obtained from the new experiment with the old one. On the basis of this prediction, a new combination of the operation parameters is calculated which is expected to yield an improved value of the COF, the separation is run at these new conditions and the procedure is repeated until maximum COF with no further improvement is eventually obtained, for which — hopefully — the optimum combination of operation parameters has been obtained (Fig. 1.22). Any combination of operation parameters can be optimised in this way and no knowledge about the nature of the chromatographic process is necessary ( black-box philosophy). Some HPLC control systems allow the simplex optimisation to run unattended. 

The main disadvantage of the simplex method consists in the laige number of experiments required to find optimal working conditions. Further, the optimisation criterion characterises the separation of the sample mixture by a single number, so that the detailed information on the separation of the individual sample components is lost and because of the high probability that the search method will slide into a region with a local maximum of the optimisation criterion, the simplex optimisation method can be expected to be fully successful only with the separations of relatively simple samples. 

The simplex method belongs to a group of optimisation methods finding the minimum of a predefined multiparameter function (error functional). The downhill simplex method of Nelder and Mead [8] requires only function... 

A reduced FORTRAN code to fit the magnetic susceptibility is available on request from the author [17]. The optimisation algorithms selected there cover the simulated annealing method combined with the simplex method. 

A. P. Wade, Optimisation of Flow Injection Analysis and Polarography by the Modified Simplex Method. Anal. Proc., 20 (1983) 523. 

Quite surprisingly, the number of papers using simplex methods to optimise analytical procedures is not so large and, therefore, our review started at 1990 (see Table 3.35). For older references on simplex applications see Grotti and Wienke et... 

B. Koklic, M. Veber and J. Zupan, Optimisation of lamp control parameters in glow discharge optical emission spectroscopy for the analysis of copper-titanium-zinc alloy using the Simplex method, J. Anal. At. Spectrom., 2003, 18(2), 157-160. 

Jensen et al. demonstrated an integrated microreactor system with online HPLC analysis to optimise the Heck reaction of 4-chlorobenzotrifluoride and 2,3-dihydrofuran (Figure 12.4). The optimisation was controlled using a Nelder-Mead Simplex method, a black-box approach, which required no a priori reaction or gradient information. They demonstrated that this process could be optimised to produce a yield of 83% after 19 experiments, each taking approximately 20 minutes including analysis time. 

Local optimisation methods include, for example, the steepest descent, Newton, quasi-Newton, conjugate gradient or simplex methods. Important concepts in local optimisation wiU be presented below. 

The second approach to optimisation is a model independent one. One of these model independent methods is the sequential simplex [24,25] used by Shek et al. [26], The method is claimed to be ideally suited for the optimisation of formulations [27] because of the relatively low number of experiments to be performed. 

Although application of chemometrics in sample preparation is very uncommon, several optimisation techniques may be used to optimise sample preparation systematically. Those techniques can roughly be divided into simultaneous and sequential methods. The main restrictions of a sequential simplex optimisation [6,7] find their origin in the complexity of the optimisation function needed. This function is a predefined function, often composed of several criteria. Such a composite criterion may lead to ambiguous results [8]. Other important disadvantages of simplex optimisation methods are that not seldom local optima are selected instead of global optima and that the number of experiments needed is not known beforehand. 

C. D. Stalikas, G. A. Pilidis and M. I. Karayannis, Determination of lead and cadmium in environmental samples by simplex optimised atomic absorption methods, J. Anal. At. Spectrom., 11(8), 1996, 595-599. 

One of the attractions of the method is that it is eminently computerisable, so that for an automated system in which the results of the simplex can be processed by a computer, which is then able to reset the apparatus for the next simplex, it is possible for a system to optimise itself. It is also, possible to detect different patterns of interaction, in the mountaineering terminology which pervades the discipline, one may detect ridges of optimisation of different peaks . 

Optimisation. This is one of the commonest applications in chemistry. How to improve a synthetic yield or a chromatographic separation Systematic methods can result in a better optimum, found more rapidly. Simplex is a classical method for optimisation (Section 2.6), although several designs such as mixture designs (Section 2.5) and central composite designs (Section 2.4) can also be employed to find optima. 

The most common, and easiest to understand, method of simplex optimisation is called the fixed sized simplex. It is best described as a series of rules. 

A weakness with the standard mediod for simplex optimisation is a dependence on the initial step size, which is defined by the initial conditions. For example, in Figure 2.37 we set a very small step size for both variables this may be fine if we are sure we are near the optimum, but otherwise a bigger triangle would reach the optimum quicker, the problem being that the bigger step size may miss the optimum altogether. Another method is called the modified simplex algorithm and allows the step size to be altered, reduced as the optimum is reached, or increased when far from the optimum. 

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