Application of Simplex Method

8/10/2019 Application of Simplex Method

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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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pplication the simplex method

68

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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Application the simplex method

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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pplication

the simplexmethod 687

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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pplication the simplex method

9

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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Application of the simplex method 691

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


12/18

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.


13/18

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


14/18

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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Received

Ma o

199

2

Accepted

June

199

3

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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).

Application of Simplex Method

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