8 NEW PRODUCT RESEARCH & DEVELOPMENT
☞
Having shown how quadratic and, even better, How then do these equations relate to the
skewed quadratic equations can be used to define and optimisation of fat compositions? Generally such
predict firstly binary and then multi-component blend optimisations are carried out to predict least-cost
properties we come to the main reason for wanting to formulations so equation (4) relates to the cost of each
predict fat compositions, the ability to use this potential component in the blend and this equation is
information to optimise the composition in some way. then minimised. Although the model could be set up to
Often there is a parameter which we wish to maximise or maximise profit instead, this generally involves a much
minimise – it may be to minimise cost or maximise larger model which includes factory process parameters
profit; it could be maximising hardness of a product or as well as just the blend optimisation module. Equation
throughput through a factory. This brings us to a (5) is the heart of the model in the sense that these
procedure known as linear programming. equations predict the physical parameters (N20, N30
etc) while the inequality terms put upper and/or lower
Table 2. Normal v Skewed Quadratic Predictions for
limits on these. Finally, equation (6) ensures that all
Blends of three Chocolate Fats (CF1, CF2 and CF3).
components are present at either zero or positive levels.
CF1 CF2 CF3 Observed Calculated Calculated
The major problem with this approach to blend
N20 Quadratic Skewed Quadratic optimisation is that it assumes linear interactions
N20 N20
between blend components; there are no quadratic
5 15 80 76.8 75.6 75.6
terms in the LP matrix. Although non-linear
5 30 65 72.0 72.1 72.7
programming is possible the equations and software
required are much more complex. However, there are
10 15 75 71.8 71.6 72.2
many instances where apparently non-linear systems can
10 30 60 66.2 67.4 68.5
be included in linear programmes. This is because there
15 15 70 70.7 66.8 68.8 are often restrictions on the ranges of components that
20 5 75 68.7 67.4 68.5
can be included and these restrictions can often be
25 15 60 64.2 59.9 61.8
linked to linearities in interactions. This is best
demonstrated by using the mixture of cocoa butter and
25 25 50 58.6 55.9 58.0
milk fat referred to earlier. Milk chocolate very rarely
30 10 60 62.0 58.5 60.3
contains more than 30 per cent milk fat so as long as the
30 20 50 57.8 54.0 58.2
portion of the cocoa butter-milk fat curve between 0 and
30 30 40 50.2 49.6 51.6 30 per cent milk fat is linear this can be used directly in
Root Mean Square Error 2.57 1.51
a linear programme. Fig. 2 shows that this is the case –
indeed the curve can be approximated to linearity
Linear programming (LP) is based on linear between 0 and 50 per cent milk fat. Once linear portions
interactions. It has been used to optimise a ‘mix’ of of all the interactions have been defined these can be
products made with a restricted availability of machines formulated into an LP matrix. A simple LP matrix
or operatives. We are using it to optimise blend containing two types of cocoa butter, two types of milk
compositions, often of confectionery fats.All LP problems fat and two types of vegetable fat is shown in Table 3.
can be reduced to a series of equations. The most
Table 3. Simple LP matrix of chocolate compositions.
important of these is the definition of the 'objective
function'. This is the parameter to be optimised –
CB1 CB2 MF1 MF2 VF1 VF2
maximised or minimised – profit or cost, for example.
a
CB1
a
CB2
a
MF1
a
MF2
a
VF1
a
VF2
≥N20
lower
b
CB1
b
CB2
b
MF1
b
MF2
b
VF1
b
VF2
≥N30
lower
Objective function Maximise or minimise:
d
CB1
d
CB2
d
MF1
d
MF2
d
VF1
d
VF2
≤N35
upper
(4) c
1
x
1
+ c
2
x
2
+ ……… + c
n
x
n
1 1 1 1 1 1 =1
c
There are then a series of limits which can be either
CB1
c
CB2
c
MF1
c
MF2
c
VF1
c
VF2
Objective Function
(minimise)
one-sided or two-sided that are put on various parameters
in the model: There are numerous ways in which such a model can
(5)
a
11
x
1
+ a
12
x
2
+ ……… + a
1n
x
n
<= b
1
then be used. Different constraints can be put on the
a
21
x
1
+ a
22
x
2
+ ……… + a
2n
x
n
<= b
2
columns in the matrix allowing formulation limits to be
…... + …… + …….. + …… <= … imposed. This enables, for example, different levels of
a
m1
x
1
+ a
m2
x
2
+ ……. + a
mn
x
n
<= b
m
vegetable fat and milk fat to be examined, constraints to
be placed on specific vegetable fats within an overall
and a final series of inequalities that ensures that 5 per cent limit, the balance between different cocoa
every component is present at either zero or a positive butter types to be varied, etc. Putting different limits on
level. the N-value parameters in each row allows different
melting profiles to be studied and their effects on cost to
(6)
x
1
>=0; x
2
>=0; …… x
n
>=0 be determined.
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