Optimisation criteria

In the previous section, the optimisation of liquid-liquid extraction with the help of mixture designs justified by the solubility theory was examined. A relation was derived between the partition coefficient and the mixture composition for liquid-liquid extraction with extraction liquids composed of three components, and a special cubic mixture model was obtained (equation (3)). 

Once a relationship between the partition coefficient and the composition of the organic phase have been found, these models can be used to build response surfaces of the partition coefficient or the recoveries of the solutes or of other criteria. 

If a substance has to be extracted from an aqueous matrix for quantitative determination, it is important to maximise its partition coefficient. The models for the partition coefficient in relation to the mixture composition can be used to estimate that mixture composition, where the partition coefficient reaches its highest value. 

The ratio of extraction of two solutes i and j is represented by the selectivity which is the ratio of the partition coefficients P, and Pj [28]  

For quantitative extraction of two solutes i and j, both P, and Pj should be maximised. If the ratio of the two partition coefficient is more important then the individual partition coefficients, a,j should be used as optimisation criterion (by definition indices of the partition coefficient are attached as such that is always smaller or equal to 1). Optimal values for a,j are those values which are equal to or which approximate unity ( ,v= I). 

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Important selection and optimisation criteria for the primary packaging may include the following ... 

IPD consists of developing alternatives rather than a unique flowsheet. The selected solution fulfils at best the optimisation criteria and the environment of constraints. 

A traditional approach for solving this problem is to apply availability and cost modelling as described by e.g NORSOK (1998), Wang Pham (2006) and Aven Jensen (1999). An optimal maintenance poHcy is determined minimi zing some optimisation criterion, for example the expected long run cost per unit of time or the total expected discounted cost. Constrained optimisation criteria have also been suggested, for example the minimisation of expected costs while some system reliability requirement(s) are satisfied, or the maximisation of system reliability when some cost require-ment(s) are satisfied see e.g., Wang and Pham (2006). 

The phosphate dosed condition that was modelled (M = 0.02, E = 30) was predicted to achieve or only slightly exceed one of the UK s optimisation criteria (Drinking Water Inspectorate, 2001) for ortho-phosphate dosing, that no more than 2% of RDT samples should exceed 10 pg/1. This dosed condition has readily been achieved in practice and, where necessary, lower values of M and E have been achieved by shghtly higher phosphate doses in order to meet the 2% RDT target (Hayes et at, 2006, 2008). 

After the sampling period, the ortho-phosphate dose can be adjusted (up or down) depending on the optimisation criteria that have been set. One of the UK s optimisation criteria was that no more than 2% of random daytime samples from consumers taps should exceed a lead concentration of 10 pg/1. 

There are almost always a number of criteria to which the formulation has to fulfil, and in the case of incorporating robustness aspects (as an optimisation criterion) into the optimisation the number of criteria is also increased. It is however almost impossible to fulfil all the criteria in the most optimal way at once. This means that a compromise has to be foimd between all criteria. A large number of methods is available to search for such a compromise variable setting. One of these methods is Pareto Optimality which will be explained and applied in this chapter. Pareto Optimality searches for a compromise between the optimisation of a certain tablet property and the optimisation of the robustness of this property. 

Now, the above procedure can also be very time-consuming when carrying out calculations on extended arrays of atoms as in crystals and other solids since convergence is often very slow and many iterations are required before the optimisation criterion is satisfied. So, these conventional methods of optimisation also became difficult on large systems. 

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. 

Fig. 1.22. Opiimisation of two operation parameters by simplex melhtxl. The dolled contour lines correspond to equal values of the optimisation criterion. ABC = original simplex. ACD = new simplex obtained by rejection of the point B with the worst value of ihe opiimisation criterion and reflecting the original simplex in the opposite field O = the combination of the iwo optimised operation parameters for the highest (optimal) value of the optimisation criterion.

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 maximnm average dissipation of specific kinetic energy of turbulisation e bas been selected as the optimisation criterion for the geometry of a tubular turbulent... 

An optimisation criterion is specified, for example the total expected discounted costs. 

The cost parameters are used as input to the optimisation criterion. The most common criteria adopted are the expected discounted cost and the expected long run cost per imit of time. These criteria can be written as functions of the cost parameters introduced. For example, if a cost ks is associated with the system being in state s per unit of time, the total expected cost in a certain interval of time J equals ... 

In practice, the ortho-phosphate dose required is also a function of the percentage reduction in plumbosolvency needed to achieve the optimisation criterion, which is strongly influenced by the extent of occurrence of lead pipes within the water supply system. In the Wales (UK) case smdy (Hayes et al., 2008), the required percentage reduction in plumbosolvency and the required average ortho-phosphate dose was established for a total of 29 dosing schemes that were subject to regulatory control, plus 9 others not subject to regulatory control. 

The argument for this criterion is very simple Who would not want to conduct a project so as to get the greatest revenue for the same cost Or, conversely, who would not want to choose the least cost approach to obtaining the same revenue Consequently, maximising the return on investment (ROI) is the common purpose of all engineering projects, and it becomes the universal optimisation criterion. 

Materials suitable as filter aids include diatomaceous earth, expanded perilitic rock, asbestos, ceUulose, nonactivated carbon, ashes, ground chalk, or mixtures of those materials. The amount of body feed is subject to optimisa tion, and the criterion for the optimisa tion depends on the purpose of the filtration. Maximum yield of filtrate per unit mass of filter aid is probably most common but longest cycle, fastest flow, or maximum utilisation of cake space are other criteria that requite a different rate of body feed addition. The tests to be carried out for such optimisation normally use laboratory or pilot-scale filters, and must include variation of the filtration parameters such as pressure or cake thickness in the optimisation. 

The application of optimisation techniques for parameter estimation requires a useful statistical criterion (e.g., least-squares). A very important criterion in non-linear parameter estimation is the likelihood or probability density function. This can be combined with an error model which allows the errors to be a function of the measured value. A simple but flexible and useful error model is used in SIMUSOLV (Steiner et al., 1986 Burt, 1989). 

Chapter 6 - There has been provided mathematical description of the processes of thermonuclear destruction in deformed polypropylene melts the aim was to use the criterion of destruction estimation in modelling and optimising the processing of polypropylene into products. 

The three robustness criteria that are explained here (Weighted-Jones (WJ), Projected-Variance (PV) and Robustness-Coefficient (RC)) describe all three the robustness of a certain mixture composition in direct relation to the response to be optimised. All three express the concept of robustness into a numerical value that can be calculated for each mixture setting (composition) of interest. So each of the criteria can be calculated as a function of the mixture composition and belongs directly to a certain response of interest. In this way a robustness criterion can be dealt with in a normal way in a mixture optimisation strategy. 

So far this approach is analogous to most of the simultaneous optimisation methods. However, the optimisation is not continued by preselecting desired values for any criterion to construct contour plots (Figure 4.13 and 4.14), or to search for acceptable solutions [29]. 

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. 

Mixture designs [13-18] are used for the optimisation of the composition of a mixture. They allow the construction of a response surface (i.e. a model) of a criterion from a relatively small number of preselected experiments. Levels of all variables cannot be chosen arbitrarily since the fractions % of the components add up to unity (for n components 0 <, < 1 + +. .. + q>= 1). Once the model function is found to be... 

Summarising, to optimise the partition coefficient P of a solute i, P, should be maximised by mixing three solvents in the correct proportions. The use of mixture design statistical techniques with the natural logarithm of the partition coefficient as response criterion is a valid way to achieve this. 

Dimensionality (or complexity) of the regression model refers to the number of factors included in the regression model. This is probably one of the most critical parameters that have to be optimised in order to obtain good predictions. Unfortunately, there is no unique criterion on how to set it. In this introductory book, only the most common procedure is explained in detail. The reader should be aware, however, that it is just a procedure, not the procedure. We will also mention two other recent possibilities that, in our experience, yield good results and are intuitive. 

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