Chem. Anal. (H1clrsaw), 38, 681 (1993) REVIEW

Application of the Simplex Method for Optimizatlonof the Analytical Methods

by C. Rozycki

Department ofFundamentals ofChemistry, Institute ofChemistry,Scientific and Didactic Centre of Warsaw 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 itsapplication in various branches of analytical chemistry,

W artykule dokonano przegladu literatury dotyczacej optymallzac]! metoda simplekso­wa i jcj zastosowania w roznych dziedzinach chemii an~n:t:ycznej.

Optimization of a chemical system consists in sucDa selection of the system-COli­trolling variables (parameters or factors, e.g. temperature; concentration, plf) whichenable a certain state-dependent variable y to achieve the most beneficial value withinthe limitations of the attainable modifications ofthe.system.

In such a case a model of the chemical system may be represented as a functionof many variables. The rcsponse y is then a value which is a characteristic of thesystem. It depends on the values of the independent variables:

y %: j{x VX2, ... , XII) (1)

Examples of optimization arc e.g, maximization of the yield of a chemicalreaction, height of an analytical signal, or minimization of an impurity component inan 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 factorwhich can give the most profitable result of the experiment. Such a method is betterthan a random search for optimum set of the factors, but other available methods canprovide more information with less labour consumption. Such a method is the Box

682 C. Rozycki

and Wilson, steepest ascent technique [1] described among others by Nalimov andChemova [2]. Various optimization methods have been described by Koehler [3]. Forthe sake of the smallest number of experiments needed and the simplicity of calcu­lations the best, method, used in chemical studies, is the one involving geometricsolids referred to as simplexes. The theory of the simplex method has been developedby Spendley et al. [4]. Literature data show that the simplex method is now the mostwidely used optimization method in analytical chemistry. Deming and Morgan [5]have discussed the bases of experimental design and quoted a bibliography of 189papers dealing with the simplex method. Moore [6] has found that 300 papers ofchemical application of the simplex method had been abstracted in Chemical Ab­stracts throughout the period 1966-1985. About 25 %ofthosepapers were concernedwith analytical chemistry. Among the analytical papers 40 % were devoted tochromatography and 15 % - to emission spectrometry. Brown-er al. [7] have noticedthat Chemical Abstracts recorded 27 papers dealing with the simplex methodthroughout the.period January 1976 - October 1979, 984 papers within January 1988- November 1989, and 1078 papers within December 1989 - November 1991. TIeattempt of the present review is to present the simplex method and its application inanalytical chemistry.

Search for optimum

Every system reacts to changes in the value of the factors (Xi) by changing thevalue of y (sometimes reffered to as the response) correspoding to the given set ofvalues ofthe factors. A sufficiently large set of responses forms the so-called responsesurface. If the number of factors is n,the response surface is (n+1)-dimensional. Sucha surface has often an extremum, which may be a point or an area. Various kinds ofthe 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 (factors) maybe regarded as coordinate axes thatform the so-called factor space, which is n-dimen­sional 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) thatmaximize or minimize the response. The definition and the study of a function givenby the relationship (1) may proceed in three steps. The first step consists in findingthe number and the kinds of independent variables Xi' In the second step the valuesof independent variables determining the position of optimum of the function are tobe found, and as the third step the relationship characterizing the response surfacenear the optimum is to be found. Of course, it is not always possible to distinguishthe three steps in a particular chemical study. The second step, which is an optimiza­tion step, is often done by the simplex method.

The simplex method

Deming and Morgan [9, 10] refer to the simplex as a geometric figure definedby a number of points higher by one compared with. the number of factors (ordimensions of the, factor space). In the two-dimensionai factor space the simplex isa triangle. and in the three-dimensional space it is a tetrahedron. In a similiar way it

Application ofthe simplex method... 683

is possible to define simplexes in multidimensional spaces as convex hyperpolyhe­dra. The simplex vertex coordinates correspond to values of the factors (or parame­ters) Xh X2, ••• , Xm for which an appropriate experiment may be performed.

The simplex method of optimization and suitable examples of its application havebeen described in a number of papers [2, 9-19]. One can find there the basic principlesof 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 experimentsperformed for the factor values corresponding to the simplex vertices. After havingcompleted experiments for all the simplex vertices the experimenter discards thevertex corresponding to the worst experimental result. The rejected vertex is nowreplaced by another one, which is its symmetrical reflection with respect to the planepassing through the other simplex vertices. By multiple repetition of that operationthe simplex shifts gradually to the part of factor space in which the results of theexperiments improve step by step. The rules of such a movement guarantee that evenif for a new vertex the corresponding result is worse than that corresponding to thediscarded one, the movement of the simplex toward the space of optimal resultscontinues. The advantage of the simplex method arises from the fact, that the decisionon further step of the simplex shift is taken after each experiment, whereas in otheroptimizing methods a greater number of experiments are performed before such adecision can be taken.

There is always a possibility, that the optimum found is a local optimum. It isimpossible to establish the global optimum without knowing the functional relation­ship (1). An optimum is probably the global one [20] if another search beginningfrom a different region of variables gives either the identical optimum position orsomething very close to it. Luand Huang [21] have described a procedure that enablesto avoid the breaks in searching within a local optimum.

The simplex method for searching has, however, some disadvantages [22]. Onlyin 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 decidewhether a given result represents an optimum, or is only a vertex, for which theresponse is better than for other vertices. In its primary version the simplex methoddid not allow for acceleration of the search of optimum because of the constant sizeof the simplex. It would be more reasonable to use a large simplex in the initial stageof the search to have a possibility of quick movement in the factor space, and todispose a smaller simplex in the final stage for more precise localization of theoptimum. The use of a simplex of variable size might allow to avoid that inconveni­ence.

Modificaton of the simplex method

Modifications introduced to the simplex method have enabled to increase theefficiency 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.

684 C. Rozycki

The contraction of the simplex involves some disadvantages: the volume of thesimplex is contracted and might give rise to premature convergence in the presenceof an error [22]. For that reason Ernest proposed, instead of contracting the simplex,to shift it in such a manner, that the vertex corresponding to an optimum result fallsin the centre of a new simplex of identical dimensions as the former one [24]. Anothersolution has been proposed by King [25]: if a vertex that was formed after thecontraction has produced a worst response, instead of it the next vertex of wrongresponse should be discarded. Such a procedure was applied by Morgan and Deming[26]. Still another solution consisting in turning the simplex has been proposed byBurgess [27]. It has also been shown [28], that in some cases, where some factorshas no substantial effect on the optimized value, a prematural contraction of thesimplex or even the end of optimization may occur. It does not mean, however, thatsuch factor has an effect and that the value it has achieved is an optimum value. Indoubtful cases further experiments have to be carried out (e.g. according to ex­perimental factor design) and the regression equation obtained should be analysed.Izakov [29] has proposed another method for designation of a new vertex in caseswhere the responses for some vertices are close to one another. In such a case two ormore vertices (instead of one) are discarded at a time thus enabling acceleration ofthe simplex movement toward the optimum.

Walters and Koon [30] varied the values of coefficients determining the size ofthe simplex (contraction and expansion) and applied various initial point and sim­plexes 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 alwaysconfirmed in practice Routh et al. [31] proposed, a Super Modified Simplex (the SMSmethod). The position of a successive simplex vertex is determined from the reponsevalue of a discarded vertex, reflection of the discarded vertex, and gravity center ofthe nondiscarded vertices (centroid). The values of responses in these three pointsare used for calculating the equation of the polynomial of the second order (aparabola). After having found the extremum of that polynomial for the range ofindependent variable values extrapolated outside the discarded vertex and its reflec­tion, it is possible to determine the position of the new vertex. The new simplex vertexis either a point corresponding to an extremum (inside the range of variables underconsideration) or at a border of the range of variables. The interval of extrapolationof the range of variables is chosen depending on the value of the first derivative ofthe polynomial. In the super modification proposed, the authors have foreseen alsosuitable procedures protecting from too early coming together of the simplex vertices,that might simulate attaining the optimum. In cases where the simplex becomesdisplaced outside the admissible factor space the new vertex is placed at the borderof this space.

Van der Wiel has described [32] further modifications of the SMS method, sincethe increase of difficulty of calculations involved with the modifications presents nomore problems and the economy of time due to decrease of the number of experimentsneeded is of primary importance. He has proposed three procedures for improvingthe SMS method. They were based on finding the new vertex by either adjusting theGauss curve to three response values: the worst vertex, the centroid, and the last

Application ofthe simplex method... 685

vertex, or by the use of the weighted method for calculation of coordinates of the newvertex, or by finally calculating the response for the new vertex instead ofperformingan experiment. Still another modification of the MS method has been proposed byRyan et al. [33]. In this method the new simplex vertex is determined from thediscarded vertex and the so-called weighted centroid. The position of the weightedcentroid depends on the interrelation of differences of response in individual simplexvertices and in the discarded vertex. To avoid a possible occurrence of simplex"degeneration" into an unidimensional simplex (only one variable influences sub­stantially the responses) two versions of the procedure have been proposed.

Also Betteridge et al. [34] have proposed two modified algorithms for searchingthe optimum by the simplex method and have verified them for selected mathematicalfunctions and for analytical methods. In. these algorithms the position of the newsimplex vertex is determined by means of the weighted centroid and the Lagrangeinterpolation. A method proposed by Routh et ale [31] has been modified [35] bygiving up the experiment in the simplex centroid and replacing its result by the meanof non-discarded simplex vertices; criteria enabling the comparison of differentversions of simplex optimization have also been proposed. Ilinko and Katsev [36]also determined the position of the new simplex vertex from the weighted centroidand 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 equilibriumbefore measurement is very long; in order to reduce the time of studies theyrecommend to carry out experiments for several vertices at the same time.

King and Deming [38] have described an optimization method called UNIPLEXwhich 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 number ofvariables andthe size of the search space. In the case of many variables the convergency of thismodification is higher than that of the NeIder-Mead method (but not for complicatedresponse surfaces). -

For more rapid attainment of the optimum and avoiding premature diminition ofthe simplex in the Nelder-Mead method, it has been proposed that the whole simplexis shifted parallelly [40].

Modifications of the simplex method have also been described in papers [41,42].The described modifications have been compared [33, 34] by simulating the

results of experiments. It has been shown, that they allow to reduce considerably thenumber of experiments needed to achieve the optimum. The progress of the optimi­zation process depends, however, also on the position of the starting simplex, theshape of the response surface, and the aim of the optimization (attainment of optimumarea or localization of the optimum position). Various modifications of the simplexmethod have been compared in [35]. The conclusions from that comparison are notunivocal: the rate of attaining the optimum of the given function depends on thealgorithm applied and the response surface. Gustavsson et al. [43-45] have comparedvarious modifications of the simplex methodfor simulated experimental results, butalso in this case it is difficult to say, which of the modifications considered is the best.It seems even, that in some cases theyhave no priority over the MS method.

686 C. Rozycki

In a number of works [46-58] the simplex optimization has been compared withother optimizing methods. As shown in [47], optimization of the spectrophotometricmethod by flow injection procedure with four variables required 88 measurements atseparate treatment of each variable, and 34 measurements with the simplex optimi­zation. For five variables the corresponding numbers are 168 and 37. Optimumconditions for chromatographic determination of carboxylic acids [55] were identicalin the case of the simplex method and the central composite design (in the latter casethe greater cost of labour gave also a mathematical description of the responsesurface). The grid and the simplex methods have been compared in [56]. Fora numberof variables lower than 4 the grid method has been recommended, since it enables,i.a., a graphical representation of the response surface. A comparison has also beenmade [59, 60] between the simplex method and the Powell method. Although in thatcase (two factors) the Powell method needed less experiments, no definite statementin favour of one or another method has been made. Five different optimizationmethods have been compared [58] for simulated data: the genetic algoritlnn wasbetter than other methods in the case, where the response surface comprised the globalmaximum, two large local maxima, and some smaller local maxima. For such casesKalivas [61, 62] has proposed to effect optimization by the simulated annealingmethod.

The history of the simplex optimization and relationships between variousmodifications of the method have been described by Betteridge et al. [34].

Realization of" the simplex method

Numerous papers [4, 12, 17, 34,52, 63,64] include a flow diagram showing thelogic of simplex method. Berridge [64] has discussed realization of the simplexmethod 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 searching the extrema of mathematicalfunctions by the simplex method. An algorithm for rapid calculation of a new simplexvertex in cases of large number of factors (about 60) has been described [72]. In themarket there are offers of programs assisting optimization by the simplex method,and even special equipment adapted for simplex optimizing of chromatographiccolumns [73]. A special spreadsheet [30] is useful in performing calculations by thesimplex method.

King et al. [74] have discussed the difficulties and the errors occurring in thecourse of optimization by the simplex method.

Combining the simplex method with the factor design permits to reduce thenumber 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 onthe problem formulation. This can be, for example, the yield of reaction, absorbance,stability of solution. Sometimes the experiments provide joint information on severalQuantities. In such cases the most important Quantity should be optimized. All the

Application ofthe simplex method... 687

other quantities may serve, if this is needed, for correcting the position of the optimumwith respect to the position determined only for the main quantity optimized. Amethod for simultaneous optimization of several quantities has recently been pro­posed [77, 78]. The criteria applied in simultaneous optimization of several featuresof 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 acriterion evaluating the quality of the fitting (the criterion of the nonlinear leastsquares method).

The course of the simplex optimization depends [78] on the choice of theoptimized value.

Selecting the factors

To avoid excessive complication of experiments only the most important factorsshould be tested. The importance of a factor is determined by comparing the changesin the response caused by a change in level of each of the factors prior to theknowledge of the system or upon preliminary experiments. The selection depends onexperience of the experimenter or on the results of preliminary experiments. ButDeming and Morgan [9, 10] did not find any disadvantageous effect of includingfactors of smaller importance on the movement of the simplex, although they canpossibly lead to premature diminishing of the simplex in modification of the simplexmethod [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 planningscreening experiments [2, 11]. Examples of such use of factorial planning are givenin [89-91]. The estimation of the effect of a given factor on the results depends alsoon the range of its values taken for the tests. Sometimes, if the range has beenimproperly selected, it may lead to omitting some important factors, as the resultsare close to each other. For that reason it is usually more disadventageous to includein the study some less important factors than to neglect an important one [92]. Thereis a possibility of increasing the number of factors at any stage of the optimizationprocess [2, 11].

The amount of the component determined and the volume of the analysed solutioncannot serve as the factors. It was shown [22, 93] that that condition had not beensatisfied in some works.

Selecting the range of the factors

It is important to select for each factor an appropriate difference of values (stepsize) to be accepted in individual vertices of the initial simplex. The selection is madearbitrarily but it is better to do it in such a way that the effect of each factor on theresponse value is similar to each other. Otherwise an apparent decrease of the numberof important factors may occur. It has been proposed [92] to select a step size that isinversely proportional to the expected value of its influence on the response.

It is advisable tobegin the search for an optimum with a large simplex (large stepsize of individual factors), as the effect of the factor should then exceed the value of

688 c. Rozycki

the experimental error [92]. A small effect of one of the factors, as compared withthat of the other factors, may arise from selection of its value near to the optimumsearched, independence of the system of that factor, or too small difference of valuesof that factor in simplex vertices.

In the literature on the simplex method there are two ways of determining thevalue of the factors. The most frequently applied method consists in using variablesdetermined in physical units, such as °C, Pa, or units of concentration. In anothermethod the values of the factors are expressed as normalized values. This system iseasier 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 anynumber of the factors. The normalized values can easily be scaled for valuesexpressed in natural units.

Constraints of the simplex method

The response surface is confined to such boundaries of variables, which resultfrom physico-chemical conditions, e.g. aggregation state, concentrations (within therange of solubility), etc. The admissible range of the factors may be defined as theexperimental region. If the vertex of a simplex moves outside this region therealization of the experiment becomes impossible. The simplest solution to thisproblem is to assign a very bad result to the unrealizable experiment and to continuethe search for the optimum. An alternative procedure to be used in cases wheresimplex 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 vertexshifted to the border of the region of variables. The usability of such a procedure hasbeen 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 individualfactors and from the initial point selected in the factor space.

Yarboro and Deming [92] have discussed, i.a., the problems connected withdetermination of the size of the initial simplex. It depends on the expected results ofthe experiments corresponding to particular vertices of the selected simplex.

End of search

The search for optimum by the simplex method ends after a certain value of anaccepted criterion has been reached (e.g. the range of values of individual variablesdiffers less than 1 % of the range in the initial simplex; the yield of the reactionreaches a value considered to be optimim by the experimenter; the variance of themeasurements for simplex vertices becomes equal to the variance of the measure­ments [10]). In the work [40] the search for a minimum was completed when thevalue of the response in three successive simplexes was lower than a predeterminedvalue. In the work [90] the search for optimum was ended when the differences in

Application ofthe simplex method..~ 689

response in vertices of the final simplex were small and one of the vertices occurredin five successive simplexes.

An algorithm has been proposed [94] for controlling the shape of the simplex (infact its symmetry) to avoid its premature contraction and thus ending the search forthe 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, 96-98] applied the factorial experiment design and canonicalanalysis of the regression equation for description of the surface response in theoptimum area and for more precise localization of the. optimum of the analysedsystem [2, 11]. The reader can also find a description of the transition from a set ofsimplex vertices to factorial experiment design enabling the determination of thesecond order regression equation [4]. In this way it is possible to acquire thedescription of the surface response in the form of a regression equation and astatistical analysis of this equation.

Applications

The following review of applications of the simplex method concerns not onlythe determining of optimum conditions for performninganalyses and measunnents,but also the selection of parameters that describe the functional relationships, solvingsystems of equations, and other problems.

Turoff and Deming [96] have described the optimization of the extraction methodof isolation of iron (III) by means of hexafluoroacetyloacetone and tri-a-butyl phos­phate for four variables. After having defined the optimum, they have achieved thedescription 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 todetermine the optimum conditions of copper extraction with acetylacetone and8-hydroxyquinoline (3 factors were optimized) [99].

Harper et al. have determined optimum conditions for an ultrasonic method ofseparation of 13 metals from atmospheric dust deposited on a filter [100].

Michalowskiet al. have used the simplex method for optimization of gravimetricdetermination of zinc in the form of 8-hydroxyquinoline complex [101].

Meuss et al. applied the simplex method for optimization of the conditions ofzinc titration with potassium ferrocyanide [102]; the conditions thus establishedenabled for more precise determination of zinc than other variants of the titrationmethod. The simplex method was used by Aggeryd and Olin to determine the endpoint of titration [103]. Using the relationship between the titrant volume and theconcentration of H+ cations they have determined the experimental parameter 11-, thedissociation constantKw and the titrant volume in the endpoint Ve. This method wasalso used for determining the number of carboxymethyl groups per glucose unit incarboxymethylcellulose.

The simplex method was applied for determining the equivalence point ofsigmoidal and segment titration curves [86].

690 c. Rozycki

Booksh et al. have described the use of the Monte Carlo method and simplexoptimization for forecasting the precision of results and selection of points ofpotentiometric curve for determining the equivalent mass with minimum error [104].

Hanatey et al. [105] have proposed that the simplex method is applied fordetermining the mechanism of the electrochemical process. Wade described theoptimization of polarographic methods [106]. The work [107] has been devoted tooptimization of the amperometric determination of glucose by the flow injectionmethod. The working conditions of enzymatic electrodes were optimized [108, 109],and the use of the simplex method for evaluation of voltammetric curve parametershave been described [54]. The simplex method of optimization has been applied tononlinear calibration of ion selective electrode array applied for determination ofNa(I), K(I), and Ca(ll) [110, 111]. The method was also applied [112] for determina­tion of the standard rate constant and the charge transfer coefficient in the case 'ofquasi-reversible electron transfer in an electrode process.

The simplex method was applied [40] for identification and determination ofcomponents of mixtures on the basis of UV-VIS spectra by comparing the obtainedspectrum with spectra from data base containing La. spectra of the components(dyestuffs and drugs) likely to occur in the mixture.

Vanroelen et al. [90] have optimized the determination of phosphates via mo­lybdenum blue. Basing on an experimental design of the type 33, (three factors andthree levels; 27 experiments repeated three times) they have identified the importantfactors, and determined their interaction and approximate range of the optimumconditions. Then they applied the simplex method (3 factors, 19 experiments) andobtained an about five-fold increase of absorbance.

Spectrophotometric determination 0 f phosphate by the flow injection method wasoptimized by Janse et al. [89], and Vacha and Strouhal applied the method foroptimizing the determination of samarium with chlorophosphonazo III [113]. Bette­ridge et al. applied the simplex method for optimization of the absorbance measuredfor the reaction of PAR with the Mn04 anion, for 4 factors [34], for spectrophotome­tric determination of isoprenaline [47], and for extraction and spectrophotometricdetermination 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 Chroma­zurol S [37], cholesterol in blood plasma [10], dibenzyl sulfoxide [88], and formal­dehyde (with chromotropic acid) [93]. Kleeman and Bailey have determined, by thesimplex method, the conditions for maximum absorption by hydrocortizone solu­tions (5 factors) [114].

The simplex method was applied for simultaneous determination of organiccomplexes of: La, Pr, Nd, Ce, and Sm (VIS spectrum) [115], and of organic com­pounds (UV-VIS spectrum) [40].

Leggett [48] has described the use of simplex method and the least squaresmethod for determining the composition of a mixture of indicators by solving asystem of equations based on spectrophotometric measurements.

Wilx and Brown applied the simplex optimization of the Kalman filter fordetermination of a known component in presence on unknown ones (or with a matrixeffect) from an UV or VIS spectra [116].

Application ofthe simplex method... 691

The simplex method was applied for optimization of fluorimetric determinationof aluminium [117].

The simplex method was utilized [56, 60, 106,109, 118, 119] for establishing thedetermination 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 kineticmethod of determination of Mo(VI) [126] and Cu(II) [127] has been discussed. Theparameters 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 determinationof Co(II) and Ag(I) by chemiluminescence in presence of gallic acid and HZ02 [129];3 variables were optimized. Guo described the use of the. simplex method fordetermining the optimum conditions of chemiluminescence method [77].

Mauro and Delaney [130] have described a method for identification of thecomponents of an IR spectrum using it. simplex optimizationIfor an unresolvedchromatographic peak).'

In an extensive work, Morgan and Deming have shown the possibility of thesimplex method in optimization of the peak resolution in gas chromatography [26].They have analysed the effect of two factors: temperature and gas flow rate (withoutand with a 30 min limit for the separation time for two-; three-, and five-componentmixtures of octane isomers. In the latter case the optimum area has been attained inthe 21st experiment. The optimum area has been described with the use of the secondorder regression equations determined on the basis of the fractional design of factorialexperiments of the type 32 (two factors and 3 levels). In the work [83], a descriptionhas 'been given of the use of a joint criterion for evaluation of chromatograms (basingon the extent of separation, number of peaks, and duration of the analysis) in simplexoptimization. Another criterion for evaluation ofgas chromatograms has been dis­cussed in [84]. An additional reduction of the number of experiments has beenachieved [75, 76] by simultaneous use of the factorial design and the. simplexoptimization for separation of a mixture of ten components. The application of thesimplex method togas chromatography has been described in papers [84, 131-134].

The application of the simplex optimization to HPLC separations has beendescribed 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 liquidchromatography.

The simplex optimization has been applied for chromatographir studies of fruitjuices [143], scent compounds [144], phospholipids [142], plan' extracts [145],amino acids [80], 12 polychlorinated biphenyls congeners [146, 147], and othercompounds (antipyretis) [82]. The paper [52] presents the elaboration on the separ­ation of nucleotides by adsorption chromatography or by reversed-phase partitionchromatography. Carboxylic acids were determined in wine [55] on the basis of thesum of the peak surfaces under optimum conditions found by the simplex method or

692 C. Rozycki

by the factorial design. Use of the factorial design followed by the simplex methodcan reduce (76] the number of experiments needed to achieve the optimum (ascompared with the simplex method alone). Thus, in the paper [148], the factorialdesign.was used for selecting the variables, for which the conditions of determinationof polycyclic aromatic hydrocarbons by gas chromatography were then determinedby 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 andBoenicke have described the optimization criterion applied [85].

The separation of ion mixtures on ion exchange resins has been optimized bySmits et al, {ISO]. To avoid the effect of the ammonium ion on the determination oftrace amounts of chlorides or sulfates, Balconi and Sigon [151] applied the Nelder­Meadmethod (MS) for optimization of the working conditions of the ion exchangecolumn.which depended on two variables (concentrations of NaOH and NaHC03) .

The simplex method was applied for optimizing the separation of Cl-, F", N03',SO~~.Olfthe ion exchange resins [152].

The PREOPT program, which is described in [153], permits to obtain prelimi­nary determination of the optimum conditions for chromatographic separation on thebasis of a theoretical model, the simplex method, and the data on the retention time.The program was applied to the literature data, and the results of the calculationshave to be checked experimentally.

Berridge has discussed the problems of automatic optimization of liquid chroma­tOg1:J[>:hy with particular consideration of the simplex method [73]. It has been shownthat.the rcarc available at least two automatic devices that enable the optimization bythc€;.simpJex method (TAMED, Laboratory Data Control, and SUMMIT, BruckerSpcetrospin),

1l.lre use of the simplex optimization to atomic absorption spectroscopy has beendiseussed [154].

Parker et al, have described the simplex optimization of atomic absorptiondet'qnninations for five variables [28]. The determination of arsenic and selenium intheform of hydrides by atomic emission spectroscopy was optimized by Parker et al.[911;Pycn et al. [155], and Sneddon [156]. Cullaj (Albania) optimized the workingpararuetcrs of the burner in a method of calcium determination [157]. The simplexmethod was used in the optimization ofdetermination of Co, Fe, Mn, and Ni in glassesby atomic absorption [53]. In the work [158], the Iactorial design followed by thesimplex method was used for optimization of mercury determination by the coldvapour method.

Also the conditions of determination with use of an inductively coupled plasmaemission spectrometer [35, 50, 51, 159-168] or capacitively coupled microwaveplasma [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 backgroundratio or other essential signal-influencing factors were optimized (for 2-5 factors).Thesimplex method was utilized [169] for optimization of the working conditionsof plasma source applied in atomic emission spectrometry.

Application ofthe simplex met/rod... 693

Reviews of the literature on the use of the simplex optimization inemissionspectrometry have been published by Moore [6J, Burton [142J and Golightly and Lear(the ICP-AES method) [170J.

Jablonsky et at. applied the simplex method of optimization for selection of theexcitation conditions in determinations by X-ray fluorescence [46]. The obtainedresults were compared with the excitation conditions proposed by a group of experts.Fiori et at. applied the simplex method for selecting the parameters of the overlappingGauss bands and determination of the area of the bands obtained in X-ray fluores­cence spectra [63]. Shew and Olsen combined the simulated annealing and thesimplex method for determining the parameters of the bi-cxpoucutial function de­scribing the fluorescence process [171].

Basing on a model of predicted spectrum in activation analysis, Burgess andHayumbu determined the optimum analytical conditions for four parameters: samplesize, duration of exposure, cooling time, and decay time, which determine thespectrum [1721. Davydov and Naumov optimized the activation determination ofmany elements [173].

Krause and Lou applied the simplex method for optimization of the conditionsof clinic analyses [174].

The simplex optimization was also applied in mass spectrometry [115, 176].Evans and Caruso applied the simplex optimization for elimination of nonspectros­copic interferences in the mass spectrometry involving inductively coupled plasma[177]. The simplex method was also used for determining the conditions enabling toeliminate the effect of chlorides on the results obtained in mass spectrometry [178].

Shavers et at. 1179], Leggett [12J, and Stieg (180] have proposed to include aspecial training of the simplex optimization of analytical methods (spectrophoto­metry, gas chromatography, and atomic absorption spectrometry) to the programmeof the university studies in chemistry.

Taule and Cassas [181 Jhave proposed to use the simplex method for determiningthe maximum or the minimum equilibrium concentration of a given chemical formbasing on the equilibrium constants, the analytical concentration, and pH of thesolution.

Rutledge and Ducause, basing on the simplex conception have developed amethod for determining the linear range of detectors [182].

An interesting and different group of papers are those devoted to the usc of thesimplex method for the other purposes. Some papers [183-185] deal with a possibilityof using the simplex method for selecting parameters of non-linear equations. Themethod presented in [184] has been discussed in papers [186, 187]. The work [188]compares the results obtained in selecting the parameters of the Arrhenius equationby different methods including the simplex method. This method can also be usedfor 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 inNMR spectrum; the criterion of quality of the spectrum was optimized by the simplexmethod. A solution of a similar problem with chromatographic peaks has beendescribed by Tomas and Sabate (191). Danielson and Malmquist basing on a locallinear model, have described the use ofsimplcxes to interpolation and calculation of

694 C. Rozycki

the expected values of a function of several variables [192]. Optimization bymeansof the simplex method was also applied for determination of absolute rate constansof racemization of amino acids [193].

The problems arising from the use of the simplex method for determination ofthe extremums of various functions have been discussed in several papers [67-71].

Optimization by the simplex method has also been proposed for determinationof the discrimination function in the pattern recognition (mass spectra were used fordistinguishing 11 functional groups in organic compounds) [194]. Wilkins et al.[195-197] have utilized the simplex optimization for determining the parametrs ofthe discriminant functions in classification of mass and NMR spectra by patternrecognition.

Lochmueller et al. have discussed the use of the simplex method in automaticanalytical 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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