How many measurements do we need to make of blends of two fats in order to be able make a fairly accurate prediction of other blends? Knowing only the two end-points restricts us to a linear interaction and we know that fats very rarely intermix in a totally linear fashion (Braipson-Danthine and Deroanne, 2006; Gioelli et al, 2003). This means that, generally, we need to move to some form of quadratic equation needing measurements on at least three and preferably more blends. Two fats that are commonly mixed together are cocoa butter and milk fat, the basis of milk chocolate. The solid fat contents are shown in Table 1.
If we were to take a purely linear interaction between the two end-points then we would predict the 50:50 blend to have an N20 of 45.7. This is clearly much higher than the measured value of 32.1 per cent. Indeed the root mean square (RMS) error if a linear interaction is used is 9.83 per cent. If, however, we add the observed value for the 50:50 blend to the two end points and define a quadratic interaction then the RMS error for the predictions of all blends in Table 1 falls to 4.67 per cent. Simply adding one extra measurement point has a tremendous effect on the accuracy of the predictive equation. Putting all of the data points in Table 1 into a quadratic equation reduces the RMS error further to 3.15 per cent. If, however, we plot the observed data alongside the predictions from the quadratic equation (using all the data) we see that there is still quite a discrepancy between the two (Fig.1) - the blue line shows the observed values, the red line the quadratic equation. There is clearly a 'skew' between 20 per cent and 50 per cent cocoa butter which the quadratic equation takes no account of. The normal quadratic equation takes the form:
(1) y = a1x1 + a2x2 + b12x1x2
where xn is the proportion of component n and a and b are constants.
We have found that adding a denominator to the quadratic term in this equation puts a 'skew' to the equation and predictions made with it. The skewed quadratic equation is of the form:
(2) y = a1x1 + a2x2 +
b12x1x2
---------
1 + c12x2
Not only can we use equations such as the skewed quadratic equation to predict binary blend properties we can combine three of these binary equations into a single equation to be able to predict a ternary composition. So the skewed quadratic terms from the binaries of components 1 and 2, 2 and 3, 3 and 1 can be combined to give a skewed quadratic equation for the ternary of components 1, 2 and 3:
(3) y = a1x1 + a2x2 + a3x3 +
b12x1x2 b23x2x3 b31x3x1
------- + ------- + -------
1+c12x2 1+c23x3 1+c31x1
The improvement in prediction using a skewed quadratic equation combined with the ability to predict ternary blend properties from a knowledge of only the binary equations is demonstrated in Table 2 in which the observed N20 of blends of three chocolate fats are compared with predictions made from normal and skewed quadratic equations.
Having shown how quadratic and, even better, skewed quadratic equations can be used to define and predict firstly binary and then multi-component blend properties we come to the main reason for wanting to predict fat compositions, the ability to use this information to optimise the composition in some way. Often there is a parameter which we wish to maximise or minimise - it may be to minimise cost or maximise profit; it could be maximising hardness of a product or throughput through a factory. This brings us to a procedure known as linear programming.
Linear programming (LP) is based on linear interactions. It has been used to optimise a 'mix' of products made with a restricted availability of machines or operatives. We are using it to optimise blend compositions, often of confectionery fats.All LP problems can be reduced to a series of equations. The most important of these is the definition of the 'objective function'. This is the parameter to be optimised - maximised or minimised - profit or cost, for example.
Objective function Maximise or minimise:
(4) c1x1 + c2x2 + ...... + cnxn
There are then a series of limits which can be either one-sided or two-sided that are put on various parameters in the model:
(5) a11x1 + a12x2 + ...... + a1nxn <= b1
a21x1 + a22x2 + ...... + a2nxn <= b2
...... + ...... + ...... + ...... <= ...
am1x1 + am2x2 + ...... + amnxn <= bm
and a final series of inequalities that ensures that every component is present at either zero or a positive level.
(6) x1>=0; x2>=0; ...... xn>=0
How then do these equations relate to the optimisation of fat compositions? Generally such optimisations are carried out to predict least-cost formulations so equation (4) relates to the cost of each potential component in the blend and this equation is then minimised. Although the model could be set up to maximise profit instead, this generally involves a much larger model which includes factory process parameters as well as just the blend optimisation module. Equation (5) is the heart of the model in the sense that these equations predict the physical parameters (N20, N30 etc) while the inequality terms put upper and/or lower limits on these. Finally, equation (6) ensures that all components are present at either zero or positive levels.
The major problem with this approach to blend optimisation is that it assumes linear interactions between blend components; there are no quadratic terms in the LP matrix. Although non-linear programming is possible the equations and software required are much more complex. However, there are many instances where apparently non-linear systems can be included in linear programmes. This is because there are often restrictions on the ranges of components that can be included and these restrictions can often be linked to linearities in interactions. This is best demonstrated by using the mixture of cocoa butter and milk fat referred to earlier. Milk chocolate very rarely contains more than 30 per cent milk fat so as long as the portion of the cocoa butter-milk fat curve between 0 and 30 per cent milk fat is linear this can be used directly in a linear programme. Fig. 2 shows that this is the case - indeed the curve can be approximated to linearity between 0 and 50 per cent milk fat. Once linear portions of all the interactions have been defined these can be formulated into an LP matrix. A simple LP matrix containing two types of cocoa butter, two types of milk fat and two types of vegetable fat is shown in Table 3.
There are numerous ways in which such a model can then be used. Different constraints can be put on the columns in the matrix allowing formulation limits to be imposed. This enables, for example, different levels of vegetable fat and milk fat to be examined, constraints to be placed on specific vegetable fats within an overall 5 per cent limit, the balance between different cocoa butter types to be varied, etc. Putting different limits on the N-value parameters in each row allows different melting profiles to be studied and their effects on cost to be determined.
Changing component costs determines the sensitivity of the system to economic changes.
As an example a model similar to that in Table 3 was constructed which contained cocoa butter, milk fat and three different vegetable fats - a cocoa butter improver, a standard CBE and a softer CBE.
Two optimisations were carried out using prices for these fats typical for mid-2007. In both optimisations cocoa butter was constrained to between 70 per cent and 100 per cent and milk fat to between 0 and 15 per cent. In the first optimisation no vegetable fat was permitted; in the second optimisation each vegetable fat could be used at levels between 0 and 15 per cent but there was a further limit that the total must not exceed 15 per cent. Limits were imposed on solid fat contents at different temperatures to allow a melting profile to be defined.
The optimised results from both runs are shown in Table 4.
Allowing the LP model to optimise the composition when vegetable fats are used has resulted in a blend with a fat phase cost £160/tonne less than when only cocoa butter and milk fat were permitted and one in which the solid fat content at 20°C and 25°C are significantly higher. Both aspects can be seen as improvements - one economic, the other functional.
Relating the properties of blends to their composition therefore not only allows predictions of other fat blends to be made, it also forms the basis of the very useful ability to optimise compositions using linear programming methodology.
Enter 9 or XX at www.scientistlive.com/efood
Geoff Talbot, The Fat Consultant, is based in Bedford, UK. Imro 't Zand is with Loders Croklaan, Malaysia; Kevin W Smith, Unilever Research, is based in Colworth Park, Bedfordshire, UK.
REFERENCES:
Braipson-Danthine S, Deroanne C (2006) - "Determination of Solid Fat Content (SFC) of Binary Fat Blends and Use of These Data to Predict SFC of Selected Ternary Fat Blends Containing Low-Erucic Rapeseed Oil" - JAOCS 83 (7) 571-581;
Gioelli LA, Simões, Rodrigues JN (2003) - "Crystal morphology and interactions of binary and ternary mixtures of hydrogenated fats" - J Food Eng. 57 347-355.
