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Chem. Anal.
H1clrsaw , 38, 681 (1993)
REVIEW
pplication of the implex
Method for Optimizatlon
the nalytical
Methods
by C. Rozycki
Department ofFundamentals ofChemistry Institute ofChemistry
Scientific n Didactic Centre ofWarsaw Technical University
09 430 Plock Poland
Key
words: simplex optimization, chemical analysis
A review is given of the literature on optimization
of
the simplex method
and
its
application in various branches of analytical
chemistry,
W artykule dokonano przegladu literatury
dotyczacej
optymallzac] metoda
simplekso-
wa i jc j zastosowania w roznych dziedzinach
chemii
a n ~ n t y c z n e j
Optimization of a chemical system consists in sucDa selection of the system-COli
trolling
variables (parameters or factors, e.g. temperature; concentration,
plf
which
enable a certain state dependent variabley to achieve the most beneficial
value within
the limitations of the attainable modifications ofthe.system.
In
such
a
case
a model
of
the
chemical
system
may
be represented as a function
of many variables. The rcsponse y is then a value which is a character istic of the
system.
depends
on the values of the
independent
variables:
y
:
j{x V
X2
XII (1)
Examples of optimization
arc
e g maximization of the yield of a chemical
reaction, height of an analytical signal, or minimization of an impurity component in
an analytical signal.
A classical method for selection
of
the
optimum
conditions consists in a
one fac
..
tor-at-a-time optimization
procedure
for finding,
such
a value of the given factor
which can
give the most profitable result of the experiment.
Such
a method is
better
than
a random
search
for optimum set of the factors, but other available methods can
provide more information with less
labour
consumption. Such a method is the
Box
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682
C.
Rozycki
and Wilson, steepest ascent technique [1] described among others by Nalimov and
hemova
[2].
Various optimizationmethods have been described by Koehler [3]. For
the
sake
of
the smallest number of experiments needed and the simplicity
of
calcu
lations the best, method, used in chemical studies, is the one involving geometric
solids referred to as simplexes. The theory of the simplex method has been developed
by Spendley
et al.
[4]. Literature data
show
that the simplex method is now the most
widely
used
optimization method in analytical chemistry.
Deming
and Morgan
]
have discussed the bases
of
experimental design and quoted a bibliography of
189
papers dealing with the simplex method. Moore [6] has found that 300 papers of
chemical
application
of the simplex method had
been
abstracted in
Chemical
b-
stracts
throughout the period 1966-1985.About 25
of thosepapers
were
concerned
with
analytical chemistry. mong the analytical papers
40 were
devoted to
chromatography and 15
to emission spectrometry. Brown-er
al.
[7] have noticed
that
Chemical Abstracts
recorded 27 papers dealing with the
simplex
method
throughout the.period January
1976 -
October
1979, 984
papers within January
1988
- November
1989,
and
1078
papers within December
1989 -
November
1991.
TIe
attempt of the present review is to present the simplex method and its application in
analytical chemistry.
Search for optimum
Every system reacts to changes in the value of the factors Xi) by changing the
value
of
y
sometimes reffered to as the response) correspoding to
the
given
set
of
values ofthe factors. A sufficiently large set of responses forms the so-called response
surface. the number of factors is n the response surface is n+1)-dimensional. Such
a surface has often an extremum,
which
may be a point or an area. Various kinds of
the
response surface occurring in the case of 2 variables are given by, among others,
Nalimov and Chernova [2] and by Long [8]. The independent
variables
actors) may
be regarded as coordinate axes thatform the so-called factor space,which is n-dimen
sional for n factors. Every experiment is represented by a point in the factor space.
Any optimization consists in finding the coordinates values
of
the factors) that
maximize or minimize the response.
The
definition and the study
of
a function given
by the relationship 1) may proceed in three steps. The first step consists in finding
the number and the kinds of independent variables
Xi
In the second step the values
of
independent variables determining the position of optimum
of
the function
are
to
be found, and as the third step the relationship characterizing the response surface
near the
optimum
is to be found. Of course, it is not always possible to distinguish
the three steps in a particular chemical study. The second step, which is an optimiza
tion step, is often done by the simplex method.
The
simplex
method
Deming and Morgan [9,
10]
refer to the simplex as a geometric figure defined
by
a
number of
points higher
by
one compared with. the number
of
factors or
dimensions
of
the, factor space). In the two-dimensionai factor space
the simplex
is
a triangle. and in the three-dimensional space it is a tetrahedron. In a
similiar
way it
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is possible to define simplexes in multidimensional spaces as convex hyperpolyhe-
dra. The simplex vertex coordinates correspond to values
of
the factors or parame-
ters) X X
m
for which an appropriate experiment may be performed.
The simplex method
of
optimization and suitable examples of its application have
been described in a number of papers [2, 9 19] One can find there the basic principles
of searching for an optimum by the simplex method. According to the method the
simplex is moved in the factor space depending on the results of experiments
performed for the factor values corresponding to the simplex vertices. After having
completed experiments for all the simplex vertices the experimenter discards the
vertex corresponding to the worst experimental result. The rejected vertex is now
replaced by another one, which is its symmetrical reflection with respect to the plane
passing through the other simplex vertices. By multiple repetition of that operation
the simplex shifts gradually to the part of factor space in which the results of the
experiments improve step by step. The rules of such a movement guarantee that even
if for a new vertex the corresponding result is worse than that corresponding to the
discarded one, the movement of the simplex toward the space of optimal results
continues. The advantage of the simplex method arises from the fact, that the decision
on further step of the simplex shift is taken after each experiment, whereas in other
optimizing methods a greater number of experiments are performed before such a
decision can be taken.
There is always a possibility, that the optimum found is a local optimum.
is
impossible to establish the global optimum without knowing the functional relation-
ship 1). An optimum is probably the global one [20] if another search beginning
from a different region of variables gives either the identical optimum position or
something very close to it. Luand Huang [21] have described a procedure that enables
to avoid the breaks in searching within a local optimum.
The simplex method for searching has, however, some disadvantages [22]. Only
in case of two factors the successive simplexes provide close packing of the space
surface). In the case of larger number of factors it is not always possible to decide
whether a given result represents an optimum, or is only a vertex, for which the
response is better than for other vertices. In its primary version the simplex method
did not allow for acceleration of the search of optimum because of the constant size
of the simplex. would be more reasonable to use a large simplex in the initial stage
of the search to have a possibility of quick movement in the factor space, and to
dispose a smaller simplex in the final stage for more precise localization of the
optimum. The use of
a simplex of variable size might allow to avoid that inconveni-
ence.
Modificaton of the simplex method
Modifications introduced to the simplex method have enabled to increase the
efficiency of searches for optima.
Nelder and Mead [23] have proposed a modified simplex method the MS -
Modified Simplex). The modification consists in introduction of two new operations:
expansion and contraction of the simplex.
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68 C Rozycki
The contraction of the simplex involves some disadvantages: the volume the
simplex is contracted and might give rise to premature convergence in the presence
an error [22]. For that reason Ernest proposed, instead contracting the simplex,
to shift it in such a manner, that the vertex corresponding to an optimum result falls
in the centre of a new simplex identical dimensions as the former one [24]. Another
solution has been proposed by King [25]: if a vertex that was formed after the
contraction has produced a worst response, instead of it the next vertex of wrong
response should be discarded. Such a procedure was applied by Morgan and Deming
[26]. Still another solution consisting in turning the simplex has been proposed by
Burgess [27].
has also been shown [28], that in some cases, where some factors
has no substantial effect on the optimized value, a prematural contraction the
simplex or even the end of optimization may occur.
does not mean, however, that
such factor has an effect and that the value it has achieved is an optimum value. In
doubtful cases further experiments have to be carried out e g according to ex-
perimental factor design and the regression equation obtained should be analysed.
Izakov [29] has proposed another method for designation of a new vertex in cases
where the responses for some vertices are close to one another. In such a case two or
more vertices instead of one are discarded at a time thus enabling acceleration of
the simplex movement toward the optimum.
Walters and Koon [30] varied the values of coefficients determining the size of
the simplex contraction and expansion and applied various initial point and sim-
plexes in the MS method in order to elucidate their effect on the optimization process.
After showing that some modifications of the simplex method are not always
confirmed in practice Routh et al [31] proposed, a SuperModified Simplex the SMS
method . The position of a successive simplex vertex is determined from the reponse
value of a discarded vertex, reflection of the discarded vertex, and gravity center of
the nondiscarded vertices centroid . The values of responses in these three points
are used for calculating the equation of the polynomial of the second order a
parabola . After having found the extremum of that polynomial for the range of
independent variable values extrapolated outside the discarded vertex and its reflec-
tion, it is possible to determine the position of the new vertex. The new simplex vertex
is either a point corresponding to an extremum inside the range of variables under
consideration or at a border of the range of variables. The interval of extrapolation
of the range of variables is chosen depending on the value of the first derivative of
the polynomial. In the super modification proposed, the authors have foreseen also
suitable procedures protectingfrom too early coming togetherof the simplex vertices,
that might simulate attaining the optimum. In cases where the simplex becomes
displaced outside the admissible factor space the new vertex is placed at the border
of this space.
Van der Wiel has described [32] further modifications of the SMS method, since
the increase of difficulty of calculations involved with the modifications presents no
more problems and the economy of time due to decrease of the numberof experiments
needed is of primary importance. He has proposed three procedures for improving
the SMS method. They were based on finding the new vertex by either adjusting the
Gauss curve to three response values: the worst vertex, the centroid, and the last
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8
vertex, or by the use of the weighted method for calculation of coordinates of the new
vertex, or by finally calculating the response for the new vertex instead ofperforming
an experiment. Still another modification of the MS method has been proposed by
Ryan et al [33]. In this method the new simplex vertex is determined from the
discarded vertex and the so-called weighted centroid. The position of the weighted
centroid depends on the interrelation of differences of response in individual simplex
vertices and in the discarded vertex. To avoid a possible occurrence of simplex
degeneration into an unidimensional simplex only one variable influences sub
stantially the responses) two versions of the procedure have been proposed.
Also Betteridge
et al
[34] have proposed two modified algorithms for searching
the optimumby the simplexmethod and have verified them for selected mathematical
functions and for analytical methods. In. these algorithms the position of the new
simplex vertex is determined by means of the weighted centroid and the Lagrange
interpolation. A method proposed by Routh et ale [31] has been modified [35] by
giving up the experiment in the simplex centroid and replacing its result by the mean
of non-discarded simplex vertices; criteria enabling the comparison of different
versions of simplex optimization have also been proposed. Ilinko and Katsev [36]
also determined the position of the new simplex vertex from the weighted centroid
and compared this method with the common simplex method. Cave and Forshaw [37]
have adapted the simplex method for cases, where the time of setting the equilibrium
before measurement is very long; in order to reduce the time of studies they
recommend to carry out experiments for several vertices at the same time.
King and Deming [38] have described an optimization method called UNIPLEX
which is a one-factor variant of the NeIder-Mead modification.
Shao [39] has developed a modification of the simplex method which introduces,
i a a relation between the initial size ofthe simplex and the numberofvariables and
the size of the search space. In the case of many variables the convergency of this
modification is higher than that of the NeIder-Mead method but not for complicated
response surfaces). -
For more rapid attainment of the optimum and avoiding premature diminition of
the simplex in the Nelder-Mead method,
it
has been proposed that the whole simplex
is shifted parallelly [40].
Modifications of the simplex method have also been described inpapers [41,42].
The described modifications have been compared [33, 34] by simulating the
results of experiments. thas been shown, that they allow to reduce considerably the
number of experiments needed to achieve the optimum. The progress of the optimi
zation process depends, however, also on the position of the starting simplex, the
shape of the response surface, and the aim of the optimization attainment of optimum
area or localization of the optimum position). Various modifications of the simplex
method have been compared in [35]. The conclusions from that comparison are not
univocal: the rate
of
attaining the optimum of the given function depends on the
algorithm applied and the response surface. Gustavsson et al [43-45] have compared
various modifications of the simplex method for simulated experimental results, but
also in this case it is difficult to say, which of the modifications considered is the best.
seems even, that in some cases
theyh ve
no priority over the MS method.
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ozy ki
In a number of works [46-58] the simplex optimization has been compared with
other optimizing methods. As shown in [47], optimization of the spectrophotometric
method by flow injection procedure with four variables required 88 measurements at
separate treatment of each variable, and 34 measurements with the simplex optimi
zation. For five variables the corresponding numbers are 168 and 37. Optimum
conditions for chromatographic determination of carboxylic acids [55] were identical
in the case of the simplex method and the central composite design in the latter case
the greater cost
of
labour gave also a mathematical description of the response
surface . The grid and the simplex methods have been compared in [56]. Fora number
of variables lower than 4 the grid method has been recommended, since enables,
i a
a graphical representation of the response surface. A comparison has also been
made [59, 60] between the simplex method and the Powell method. Although in that
case two factors the Powell method needed less experiments, no definite statement
in favour of one or another method has been made. Five different optimization
methods have been compared [58] for simulated data: the genetic algoritlnn was
better than othermethods in the case, where the response surface comprised the global
maximum, two large local maxima, and some smaller local maxima. For such cases
Kalivas [61, 62] has proposed to effect optimization by the simulated annealing
method.
The history
of
the simplex optimization and relationships between various
modifications of the method have been described by Betteridge
et al [ 4]
Realization o the simplex method
Numerous papers [4, 12, 17, 34 5 63 64] include a flow diagram showing the
logic of simplex method. Berridge [64] has discussed realization of the simplex
method by means of a microcomputer. This problem has been touched also in [44],
where various versions of the simplex method are compared. Monographs [65, 66]
and some papers [67-71] include programs for searchingthe extrema of mathematical
functions by the simplex method. An algorithm for rapid calculation of a new simplex
vertex in cases of large number of factors about 60 has been described [72]. In the
market there are offers of programs assisting optimization by the simplex method,
and even special equipment adapted for simplex optimizing
of
chromatographic
columns [73]. A special spreadsheet [30] is useful in performing calculations by the
simplex method.
King
et al
[74] have discussed the difficulties and the errors occurring in the
course of optimization by the simplex method.
Combining the simplex method with the factor design permits to reduce the
number of measurements needed as compared with the simplex method alone [75, 76].
Quantity to be optimized
The selection
of
the quantity to be optimized the response depends directly on
the problem formulation. This can be, for example, the yield of reaction, absorbance,
stability of solution. Sometimes the experiments provide joint information on several
Quantities. In such cases the most important Quantity should be optimized. All the
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otherquantities may serve,
if
this is needed, for correcting the position
of
the optimum
with respect to the position determined only for the main quantity optimized. A
method for simultaneous optimization of several quantities has recently been pro
posed [77, 78]. The criteria applied in simultaneous optimization of several features
of chromatograms have been discussed in a number of papers [21 52 79 85].
In fitting theoretical curves to experimental data [86] the optimized value was a
criterion evaluating the quality of the fitting the criterion of the nonlinear least
squares method .
The course of the simplex optimization depends [78] on the choice of the
optimized value.
Selecting the factors
To avoid excessive complication of experiments only the most important factors
should be tested. The importance of a factor is determined by comparing the changes
in the response caused by a change in level of each of the factors prior to the
knowledge of the system or upon preliminary experiments. The selection depends on
experience of the experimenter or on the results of preliminary experiments.
But
Deming and Morgan [9, 10] did not find any disadvantageous effect of including
factors of smaller importance on the movement of the simplex, although they can
possibly lead to premature diminishing of the simplex in modificationof the simplex
method [28].
The selection of the factors can be. done by using
the
factorial design method,
especially the fractional factor design method [87, 88], and the methods of planning
screening experiments [2, 11]. Examples of such use of factorial planning are given
in [89-91]. The estimation of the effect of a given factor on the results depends also
on the range of its values taken for the tests. Sometimes, if the range has
been
improperly selected, t may lead to omitting some important factors, as the results
are close to each other. For that reason it is usually more disadventageous to include
in the study
some
less important factors than to neglect an important one [92]. There
is a possibility of increasing the number of factors at any stage of the optimization
process [2, 11].
The amount of the component determined and the volume of the analysed solution
cannot serve as the factors.
t
was shown [22, 93] that that condition had not
been
satisfied in some works.
Selecting
the range
of
the
factors
t
is important to select for each factor an appropriate difference of values step
size to be accepted in individual vertices of the initial simplex. The selection is made
arbitrarily but it is better to do it in such a way that the effect of each factor on the
response value is similar to each other. Otherwise an apparent decrease
of
the number
of important factors may occur.
t
has been proposed [92] to select a step size that is
inversely proportional to the expected value of its influence on the response.
t is advisable
tobegin
the search for an optimum with a large simplex large step
size of individual factors , as the effect of the factor should then exceed the value of
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c. Rozycki
the experimental error [92]. A small effect of one of the factors, as compared with
that of the other factors, may arise from selection of its value near to the optimum
searched, independence of the system of that factor, or too small difference of values
of that factor in simplex vertices.
In the literature on the simplex method there are two ways
of
determining the
value
of
the factors. The most frequently applied method consists in using variables
determined in physical units, such as C, Pa, or units of concentration. In another
method the values of the factors are expressed as normalized values. This system is
easier for the purpose of presentation of the theory of the simplex method [2, 8, 11, 13].
These papers include also formulas and tables of normalized variable values for any
number of the factors. The normalized values can easily be scaled for values
expressed in natural units.
Constraints of the simplex method
The response surface is confined to such boundaries of variables, which result
from physico-chemical conditions, e.g. aggregation state, concentrations within the
range of solubility),
etc.
The admissible range of the factors may be defined as the
experimental region.
the vertex of a simplex moves outside this region the
realization of the experiment becomes impossible. The simplest solution to this
problem is to assign a very bad result to the unrealizable experiment and to continue
the search for the optimum. An alternative procedure to be used in cases where
simplex shifts outside the admissible region was described by Van der Wiel et al.
[94]. Cave has proposed [95] a procedure in which an experiment is done for a vertex
shifted to the border of the region of variables. The usability of such a procedure has
been checked using simulated results.
Initial simplex
The position of the initial simplex is determined from preliminary experiments.
The coordinates of the vertices may be calculated from the step size of individual
factors and from the initial point selected in the factor space.
Yarboro and Deming [92] have discussed, i.a. the problems connected with
determination of the size of the initial simplex.
depends on the expected results of
the experiments corresponding to particular vertices of the selected simplex.
n of search
The search for optimum by the simplex method ends after a certain value of an
accepted criterion has been reached e.g. the range of values of individual variables
differs less than 1 of the range in the initial simplex; the yield of the reaction
reaches a value considered to be optimim by the experimenter; the variance of the
measurements for simplex vertices becomes equal to the variance
of
the measure
ments [10]). In the work [40] the search for a minimum was completed when the
value
of
the response in three successive simplexes was lower than a predetermined
value. In the work [90] the search for optimum was ended when the differences in
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response in vertices of the final simplex were small and one of the vertices occurred
in five successive simplexes.
An algorithm has been proposed [94] for controlling the shape of the simplex in
fact its symmetry to avoid its premature contraction and thus ending the search for
the optimum. Other solutions of the problem has been given in [33].
Surface response
After having ended the optimization process by means of the simplex method,
some authors [26, 9 98] applied the factorial experiment design and canonical
analysis of the regression equation for description of the surface response in the
optimum area and for more precise localization of the. optimum of the analysed
system [2, 11].
The
reader can also find a description of the transition from a
set
of
simplex vertices to factorial experiment design enabling the determination of the
second order regression equation
[4].
In this way it is possible to acquire the
description of the surface response in
the
form of a regression equation and a
statistical analysis of this equation.
Applications
The following review of applications of the simplex method concerns not only
the determining of optimum conditions for performninganalyses and measunnents,
but also the selection
of
parameters that describe the functional relationships, solving
systems of equations, and other problems.
Turoffand Deming [96] have described the optimizationof the extraction method
of
isolation
of
iron III by means of hexafluoroacetyloacetone and tri-a-butyl phos
phate for four variables. After having defined the optimum, they have achieved the
description of the optimum area with a polynomial of the second order by means of
. a
composite
design. The simplex method was used by McDevitt and
Barker
to
determine the optimum conditions of copper extraction with acetylacetone and
8-hydroxyquinoline 3 factors were optimized [99].
Harper et al have determined optimum conditions for an ultrasonic method of
separation of 13 metals from atmospheric dust deposited on a filter [100].
Michalowskietal have used the simplex method for optimization
of
gravimetric
determination of zinc in the form of 8-hydroxyquinoline complex [101].
Meuss
et al
applied the simplex method for optimization of the conditions of
zinc titration with potassium ferrocyanide [102]; the conditions thus established
enabled for more precise determination of zinc than other variants of the titration
method. The simplex method was used by Aggeryd and Olin to determine the end
point
of titration [103]. Using the relationship between the titrant volume and the
concentration of H+ cations they have determined the experimental parameter 11 the
dissociation
constantK
w
and the titrant volume in the
endpoint
V
e
.
This method was
also used for determining the number of carboxymethyl groups per glucose unit in
carboxymethylcellulose.
The simplex method was applied for determining the equivalence point of
sigmoidal and segment titration curves [86].
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ozycki
Booksh et al have described the use of the Monte Carlo method an d simplex
optimization for forecasting the precision of results and selection of points of
potentiometric curve for determining the equivalent mass with
minimum error
[104].
Hanatey et al [105] have proposed that the simplex method is applied for
determining the mechanism of the electrochemical process. Wade described the
optimization of polarographic methods [106]. The work [107] has been devoted to
optimization of the amperometric determination of glucose by the flow injection
method. The working conditions of enzymatic electrodes were optimized [108, 109],
and the use of the simplex method for evaluation of voltammetric curve parameters
have
been
described [54]. The simplex method of optimization has
been
applied to
nonlinear calibration of ion selective electrode array applied for determination of
Na I , K I , and Ca ll)
[110, 111].
The method was also applied
[112]
for determina
tion
of
the standard rate constant
and
the charge transfer coefficient in the case
o f
quasi-reversible electron transfer in an electrode process.
The simplex method was applied
[40]
for identification and determination of
components
of
mixtures on the basis of UV-VIS spectra by comparing the obtained
spectrum with spectra from data base containing a spectra of the components
dyestuffs and drugs likely to occur in the mixture.
Vanroelen et al [90] have optimized the determination of phosphates via mo
lybdenum blue. Basing on an experimental
design
of the type 3
3
,
three factors and
three levels; 27 experiments repeated three times they have identified the important
factors, and determined their interaction and approximate range
of
the
optimum
conditions.
Then
they applied the simplex method 3 factors, 19 experiments and
obtained an about five-fold increase of absorbance.
Spectrophotometric determination
0
f phosphate by the flow injectionmethod wa s
optimized by Janse et al [89], and Vacha and Strouhal applied the method for
optimizing the determination of samarium with chlorophosphonazo
II I
[113]. Bette
ridge et al applied the simplex method for optimization of the absorbance measured
for the reaction of PAR with the Mn04 anion, for
4
factors
[34],
for spectrophotome
tric determination of isoprenaline [47], and for extraction and spectrophotometric
determination
of
U VI with PAN by the flow injection method, for
12
factors
[34].
The method was used for optimizing the determination of aluminium with Chroma
zurol S
[37],
cholesterol in blood plasma [10], dibenzyl sulfoxide [88], and formal
dehyde with chromotropic acid [93]. Kleeman and Bailey have determined, by the
simplex method, the conditions for maximum absorption by hydrocortizone solu
tions 5 factors [114].
The simplex method
wa s
applied for simultaneous determination of organic
complexes of: La, Pr, Nd, Ce, and Sm VIS spectrum [115], and of organic com
pounds UV-VIS spectrum
[40].
Leggett
[48]
has described the use of simplex method and the least squares
method for determining the composition
of
a mixture of indicators by solving a
system
of
equations based on spectrophotometric measurements.
Wilx and
Brown
applied the simplex optimization of the
Kalman
filter for
determination of a known component in presence on unknown ones or with a matrix
effect from an UV or VIS spectra [116].
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The simplex method was applied for optimization of fluorimetric determination
of aluminium [117].
The simplex method was utilized [56, 60, 106,109, 118, 119] for establishing the
determination conditions in flow injection analysis,
i a
of ammonium ion [59],
Fe III) and Fe II) in solutions [120], glucose [107, 108, 121], isoprenaline [34 47
122], hydroxylamine [123], chlorohexadine by turbidimetric method) [124], ni .
trogencompounds
after enzymatic reduction to ammonium ion [91], uranium VI)
[34], and tetracyclin group antibiotics [125].
The possibility of using the simplex method for optimization of the kinetic
method of determination of Mo VI) [126] and Cu II) [127] has been discussed. The
parameters of kinetic curves used in photometric determination of Mn II) and Pb II)
were also determined [128] with the use of the simplex method.
Stieg and Nieman have described the simplex optimization of the determination
of Co II) and Ag I) by chemiluminescence in presence of gallic acid and
HZ
[129];
3 variables were optimized. Guo described the use of the. simplex method for
determining the optimum conditions of chemiluminescence method [77].
Mauro and Delaney [ 130] have described a method for identification of the
components of an IR spectrum using t simplex optimizationIfor an unresolved
chromatographic peak).
In an extensive work, Morgan and Deming have shown the possibility of the
simplex method in optimization of the peak resolution in gas chromatography [26].
They have analysed the effect of two factors: temperature and gas flow rate without
and with a 30 min limit for the separation time for two-; three-, and five-component
mixtures of octane isomers. In the latter case the optimum area has been attained in
the 21st experiment. The optimum area has been described with the use of the second
order regression equations determined on the basis of the fractional design
of
factorial
experiments of the type 3
two factors and 3 levels). In the work [83], a description
has been given of the use
of
a joint criterion for evaluation of chromatograms basing
on the extent of separation, number of peaks, and duration of the analysis) in simplex
optimization. Another criterion for evaluation
of
gas chromatograms has been dis
cussed in [84]. An additional reduction of the number of experiments has been
achieved [75,
76]
by simultaneous use of the factorial design and the. simplex
optimization for separation of a mixture of ten components. The application of the
simplex method togas chromatography has been described in papers [84, 131-134].
The application of the simplex optimization to HPLC separations has b een
described in many papers [57, 64, 73 76 80 135 140]. Berridge [141] and Burton
[142] have published reviews on the use of the simplex method in high pressure liquid
chromatography.
The simplex optimization has been applied for chromatographir studies of fruit
juices [143], scent compounds [144], phospholipids [142], plan extracts [145],
amino acids [80], 12 polychlorinated biphenyls congeners [146, 147], and othe r
compounds antipyretis) [82]. The paper [52] presents the elaboration on the separ
ation
of
nucleotides by adsorption chromatography or by reversed-phase partition
chromatography. Carboxylic acids were determined in wine [55] on the basis of the
sum of the peak surfaces under optimum conditions found by the simplex method or
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692 C ozy ki
by the factorial design. Use of
the
factorial design followed by the
simplex
method
can reduce
(76]
the
number of experiments needed to
achieve
the
optimum
as
compared
with
the
simplex method
alone .
Thus, in
the paper [148],
the
factorial
design.was
used
for
selecting
the
variables,
for
which
the
conditions
of
determination
of polycyclic aromatic hydrocarbons by gas chromatography were then determined
by the-simplex method.
The-optimization in thin-layer chromatography has
been
described also [85, 149].
Blanco
applied
that method jointly
with
the factorial
design
[149],
and
Howard
and
Boenicke have described the optimization
criterion applied
[85].
The separation of ion mixtures on ion exchange resins has
been
optimized by
Smits et l
{ISO]. To avoid the
effect of
the
ammonium ion
on the determination
of
trace amounts of chlorides or sulfates, Balconi and Sigon [151] applied
the
Nelder
Meadmethod
MS
for
optimization
of
the
working
conditions
of
the ion
exchange
column.which
depended on two variables concentrations of NaOH and NaHC0
3
The
simplex method was applied for optimizing the separation of Cl-, F , N03 ,
S ~ ~ l f t h e
ion
exchange resins [152].
The PREOPT
program,
which is
described
in [153] ,
permits
to obtain
prelimi
nary
determination
of
the
optimum
conditions for chromatographic separation
on
the
basis of a theoretical model, the
simplex method,
and the data on
the
retention time.
heprogram
was applied
to the
literature
data,
and
the results
of
the
calculations
have to be checked experimentally.
Berridge has
discussed
the problems of automatic optimization
of liquid
chroma
tOg :J[>:hy with
particular consideration of
the simplex method [73]. thas been shown
that.the r
carc
available
at least
two
automatic devices that enable the optimization by
thc;.simpJex method
TAMED,
Laboratory Data
Control, and
SUMMIT, Brucker
Spcetrospin),
l lreuse of
the simplex optimization to atomic
absorption
spectroscopy has been
dis uss [154].
Parker et al
have
described the simplex
optimization
of
atomic
absorption
det qnninations for five variables [28]. The
determination
of arsenic and
selenium
in
theform of hydrides by atomic emission spectroscopy was optimized by Parker
et al
[911;Pycn
et al
[155], and
Sneddon
[156]. Cullaj
Albania optimized
the
working
pararuetcrs of the burner in a method of calcium determination [157]. The simplex
method was used in the optimization ofdetermination
of
Co, Fe, Mn, and Ni in glasses
by atomic absorption [53]. In the
work
[158] , the Iactorial design followed by the
simplex method was used for optimization
of mercury determination
by the
cold
vapour
method.
Also the conditions
of
determination
with use of
an inductively coupled
plasma
emission spectrometer [35, 50, 51, 159-168]
or
capacitively
coupled
microwave
plasma [87] were optimized by the simplex method. Pb, AI, Na or Ca were determined
[5l .ln
these
works
the
measured
signal was maximized
or
the
signal
to
background
ratio or
other
essential
signal-influencing factors
were
optimized for 2-5 factors).
Thesimplex method was uti lized [169] for optimization
of
the working conditions
of plasma source applied in atomic emission spectrometry.
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Application the simplex met rod
693
Reviews of the literature on the use of the simplex optimization inemission
spectrometry have been published by Moore [6J, Burton [142J and Golightly nd Lear
the ICP-AES method
[170J.
Jablonsky
et
t
applied the
simplex
method
of
optimization
for
selection
of
the
excitation conditions in determinations by X-ray fluorescence
[46].
The
obtained
results were compared
with
the excitation conditions
proposed
by
group of experts.
Fiori
et
t applied the simplex method for selecting the parameters of the overlapping
Gauss
bands and
determination
of
the area of the bands obtained in
X-ray
fluores
cence spectra [63]. Shew and Olsen combined the simulated annealing and the
simplex method
for
determining
the parameters
of
the bi-cxpoucutial function de
scribing the fluorescence process [171].
Basing on a model of predicted spectrum in activation analysis, Burgess
and
Hayumbu
determined the
optimum
analytical conditions for four parameters:
sample
size, duration of exposure cooling time, and decay time, which determine the
spectrum
[1721.
Davydov
and Naumov optimized the activation determination of
many elements [173].
Krause and
Lou
applied the simplex method for
optimization of
the conditions
of clinic analyses [174].
The
simplex optimization was
also applied in mass
spectrometry [115, 176].
Evans and Caruso applied the simplex optimization for elimination of nonspectros
copic interferences in the mass spectrometry involving inductively coupled plasma
[177]. The simplex method was also used for determining the conditions enabling to
eliminate the
effect of chlorides
on the results obtained in mass spectrometry
[178].
Shavers et t 1179],
Leggett
[12J, and Stieg 180] have proposed to include a
special training of the
simplex
optimization of analytical methods spectrophoto
metry, gas
chromatography and atomic
absorption
spectrometry
to the
programme
of
the university studies in chemistry.
Taule and
Cassas
[181
have proposed to use the
simplex
method for
determining
the maximum or the minimum equilibrium concentration of a given chemical form
basing on the
equilibrium
constants, the analytical concentration, and
pH of
the
solution.
Rutledge and
Ducause basing
on the simplex conception have developed a
method for determining the linear range of detectors [182].
An interesting and different group of papers are those devoted to the usc of the
simplex method for the other purposes. Some papers [183-185] deal with a possibility
of using the simplex
method
for
selecting
parameters of non-linear equations. The
method presented in [184] has been discussed in papers [186, 187]. The work [188]
compares the results
obtained
in selecting the parameters of the Arrhenius
equation
by different methods including the simplex method. This method can also be used
for finding a non-linear equation which fits best to experimental results
[189 .
Akitt
[190]
has
described
a
method
for
selecting
the parameters
of
the overlapped lines in
NMR
spectrum; the criterion of quality of the spectrum was optimized by the simplex
method. A solution of a
similar problem
with
chromatographic
peaks has
been
described by Tomas and Sabate 191 . Danielson and Malmquist basing on a local
linear model,
have
described the use
ofsimplcxes
to interpolation and calculation of
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694
C. Rozycki
the expected values of a function of several variables [192]. Optimization bymeans
of the simplex method was also applied for determination of absolute rate constans
of
racemization
of
amino acids [193].
The problems arising from the use of the simplex method for determination of
the extremums of various functions have been discussed in several papers [67-71].
Optimization by the simplex method has also been proposed for determination
of the discrimination function in the pattern recognition (mass spectra were used for
distinguishing 11 functional groups in organic compounds) [194]. Wilkins
et al.
[195-197] have utilized the simplex optimization for determining the parametrs of
the discriminant functions in classification of mass
and
NMR spectra by pattern
recognition.
Lochmueller
et al.
have discussed the use
of
the simplex method in automatic
analytical devices [198].
The simplex method enables the automatic
fOCUSSIng
of an ion beam [199].
Examples
of
the use of the simplex method for increasing the yield of a chemical
reaction are given in [200, 201].
The
simplex
method
may also be used for
optimization of
the
Kalman
filter
[116, 202]. .
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201. Bicking M. K L and Adinolfe N. A J Chromatogn Sci. 23, 348 1985); Chem. Abstr. 1 .:3 16578
{
1985).
202.Rutan S. C. and Brown S. D. Anal. Chim. Acta 167 39 1985).
Received
Ma o
199
2
Accepted
June
199
3
ADDITIONAL REFERENCES
General:
Bezegh
A
Magy. Kem. Foly 96, 522 1990); Chem. Abstr. 118, 115679a 1993).
Plonvier J. Ch., Corkan
L
A and Lindsey J. S., Chemom. Intell. Lab Syst. 17, 75 1992).
Titrimetric methods:
Lim Ho Jin, Lae Mang Ho and KimInWhan PunsokKwahak 1, 179 198,8);
Chem.Abstr.
118,
la;z3:1
8V
1993).
PIA - spectrophotometry:
SultanS. M. and Suliman E E. a Anal. Sci. 8, 841 1992).
Electrochemical methods:
Oduza C. E, Chemom. Intell. Lab. Syst. 17, 243 1992).
Chromatographic methods:
Rakotomanga S., Baillet A and Pellerin E J Pharm. Biomed.Anal. lO 587 1992). .
Palasota J. A Leonodou1 PalasotaJ. M., Chang H. and
Deming S,
N. Anal. Chim.Acta 270, 101 1 ~ 2 ) -
Haernaelaeinen M. D., Liang Y., Kvalheim a. M. and Andersson R. Ana/. Chlm.Acta 271, 101 1 9 ~ 2 ) .
)
Curve fitting:
Glab S., { oncki R. and Holona I. Analyst 117, 1671 1992).