Ž .Chemometrics and Intelligent Laboratory Systems 48 1999 81–90

Computer-assisted qualimetric optimization of analytical methods

E. Vereda-Alonso a, J.A. Perez-Hidalgo a, A. Rıos b, M. Valcarcel b,)´ ´ ´a Department of Analytical Chemistry, Faculty of Sciences, UniÕersity of Malaga, E-29071, Malaga, Spain´ ´

b Department of Analytical Chemistry, Faculty of Sciences, UniÕersity of Cordoba, E-14004, CordoÕa, Spain´

Accepted 5 February 1999

Abstract

Software that implements the SIMPLEX optimisation method and the quality requirements of laboratories working onvarious tasks was developed. The program presents the users with different optimum experimental conditions according topreselected quality criteria, these variables being subsequently subjected to robustness test. This methodology is based on thedevelopment of a special response function from the area of a polygon constructed from four main variables, viz. sensitivity,precision, cost and throughput, each of which is weighted by a ‘quality coefficient’. In this way, the principal parametersthat define overall analytical quality are integrated and weighted in order to obtain information on the quality parameters onwhich the quality policy of a given laboratory relies. The system was used to optimise the flow injection method for thedetermination of furfural with aniline. The results obtained are discussed, and the design of the apparatus and software iscommented on. q 1999 Elsevier Science B.V. All rights reserved.

Keywords: Qualimetry; Optimisation; SIMPLEX method; Response function; Furfural determination

1. Introduction

The quality of analytical results relies heavily ontwo analytical properties: accuracy and representa-tiveness. These properties in turn depend on another

Žgroup of analytical properties called basic proper-.ties such as sensitivity, selectivity and precision,

which determine the quality of the analytical resultsobtained inside the laboratory. The analytical processis also limited by seemingly less significant, acces-sory properties. These properties, however, oftenhave major practical implications. They include ex-peditiousness and cost-effectiveness, which affect

) Corresponding author. Tel.: q34-57-218614; Fax: q34-57-218606; E-mail: qa1ricaa@uco.es

laboratory productivity. Thus, the generic objectivesof present-day analytical chemistry are to obtainlarger amounts of analytical information of higherquality by using less material, time and human re-

w xsources 1 .In developing an analytical method, one may need

to adjust five or more variables in order to establishthe optimum conditions for analyses. This can be ex-tremely time-consuming if conventional univariateoptimisation is done manually; so where interactionsbetween the variables exist, one is unlikely to find the

w xtrue optimum 2 . The problem is even heavier if theoptimisation procedure must be in accordance withthe quality policy of a given laboratory.

To avoid these problems, significant efforts arebeing made in the field of computer-assisted optimi-

0169-7439r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0169-7439 99 00014-3

( )E. Vereda-Alonso et al.rChemometrics and Intelligent Laboratory Systems 48 1999 81–9082

sation; so far, optimisation algorithms have providedw xencouraging results 3–10 . Among direct optimisa-

Žtion methods, the SIMPLEX method originally re-w xported by Deming and Morgan 11 and Deming and

w x .Parker 12 in Analytical Chemistry is the mostcommon choice for analytical chemical applications.A modified version of wide acceptance among ana-

w xlytical chemists 13,14 was devised by Spendley etw x w xal. 15 and slightly altered by Nelder and Mead 16 .

This useful algorithm is known as the modified SIM-Ž .PLEX method MSM . However, any improvements

rely on correct selection of the response function byassessing different alternatives and choosing thatwhich best meets the desired quality requirements orby re-weighting the variables concerned, so resultingin experimental work-load limitations.

w xRecently 17 , the feasibility of coupling themathematical model with optimisation algorithmswas evaluated by following a previously studied re-sponse function. In this work, a methodology that cancircumvent the abovementioned shortcomings wasdeveloped; it is based on coupling a mathematicalmodel descriptive of the response function to a SIM-PLEX algorithm via computer software. The pro-gram presents the user with different optimal experi-mental conditions according to quality criteria, thevariables being subsequently subjected to a robust-ness test. This software was used to optimise the de-

w xtermination of furfural reported by Alonso et al. 18 ,which is moderately complex and uses a very com-monplace FI manifold. We selected a FI system be-cause it is flexible, easy to configure, responsive andmimetic with many other types of system.

2. Experimental

2.1. Reagents

Analytical reagent grade chemicals were usedthroughout. Aniline was purified by distillation at183–1858C and stored refrigerated. Before use, ani-line was checked to be still colourless.

ŽThe aniline reagent was prepared as follows for.instance, for 30% aniline : 30 ml of distilled aniline

was dissolved in 60 ml of glacial acetic acid and di-luted to 100 ml with doubly deionised water.

Furfural was purified by passage through a silicagel column. A 1% wrv stock standard solution wasprepared from fresh purified furfural and stored re-frigerated in the dark. Working standard solutionswere prepared daily by appropriate dilution.

2.2. Apparatus

A Pye Unicam 8625 spectrophotometer connectedto a Perkin-Elmer R100 recorder was used for detec-tion. In addition, the FI manifold comprised a Shar-lau four-channel peristaltic pump with siliconepropulsion tubes, a Rheodyne-50 injection valve anda Hellma QS-1 cm flow-cell. The reactor coil wasmade from PTFE tubing of 0.5 mm i.d.

2.3. Manifold

A typical two-channel manifold was used for thedetermination of furfural. The sample was injectedinto a distilled water carrier that was subsequentlymerged with the reagent solution stream and drivento a reactor coil to react prior to the photometric de-tector.

2.4. Software

The system software, written in Cqq, is menu-driven and interactive. It includes a modified SIM-PLEX optimisation procedure, the mathematicalmodel of our response function that enables auto-

Žmated optimisation of analytical methods includingall the parameters that define the overall analytical

.quality, in weighted form , and robustness test foroptimise variables.

2.5. Mathematical model for the response functionand algorithms

The response function was chosen in accordanceŽwith two basic analytical properties viz. sensitivity

.and precision , which, as not tested earlier, deter-mine the quality of the analytical results obtained in-

Žside the laboratory, and two accessory properties viz..cost and throughput , which affect laboratory produc-

tivity. Each of these properties was represented as theaxis of a coordinate system to construct a polygon

( )E. Vereda-Alonso et al.rChemometrics and Intelligent Laboratory Systems 48 1999 81–90 83

Fig. 1. Graphical representation of the response function.

Ž .Fig. 1 , the area of which determine our responsefunction

sSqcC aAqrRŽ . Ž .As

20,000

As can be seen, the response function is a weightedlinear combination of variables that can be used asguidance to different compromise quality require-ments. S, C, A and R are weighting coefficientsŽ .called ‘quality coefficients’ that can be varied be-tween 0 and 100%. S, C, A and R are the above de-scribed analytical properties, each being calculated as

Ža function of two other variables except throughput. Žwhich depends on a single variable : Ss f slope of. Žcalibration curve, limit of detection ; Cs f sample. Žconsumption, reagent consumption ; A s f relative

standard deviation of the slope of the calibrationcurve, relative standard deviation of the determina-

. Ž .tion ; and Rs f time of analysis .Each value of the different variables is previously

normalised by using the following expression:

R yRŽ .max expRs

R yRŽ .max min

where R is the experimental value of the variableexp

measured, and R and R the correspondingmin max

minimum and maximum experimental values, re-spectively.

All variables were minimised; exceptionally, theslope of the calibration curve was maximised as:

S ySŽ .exp minR s .slope S ySŽ .max min

The algorithm used for the SIMPLEX optimisa-w xtion was based on the MSM of Nelder and Mead 16 .

A double convergence criterion was used in theSIMPLEX algorithm. First, the standard deviationbetween the values of the response function, called Q,should be less than a preset value, fixed by the userat the beginning of the optimisation procedure. In this

Žstudy, the value was fixed at 0.20 this can bechanged from the Functions Menu, simply by choos-

.ing a ‘‘New value for Q’’ . Second, the differencesbetween the values of the last response functions werenot always positive or negative, at least one of themshould be of different sign than the others.

2.6. Robustness studies

Robustness is a measure of resistance of the ana-Žlytical response to slight changes or significant

changes but always within the optimum region for.every factorrvariable in the experimental variables.

The robustness test employed here was the Youden–Steiner test, which involves constructing a table from

Žfour, eight or 12 experiments for 1–3, 4–7 and 8–11.experimental variables, respectively by using differ-

ent combinations of a high and a low value of eachexperimental variable in such a way that the arith-metic mean of the results of two, four or six experi-ments is the value of the response function at the highor low value of each experimental variable while theothers cancel one another. Thus, the robustness ofeach variable is determined as the difference be-tween each upper and lower case quantity.

The high and low value of each variable was givenby the SIMPLEX optimisation; the two were the lim-its of the ranges optimised by the SIMPLEX algo-rithm.

2.7. Procedure

The values of the response function were evalu-ated by constructing a calibration curve from five

Table 1Starting points used in the optimisation experiments

y1Ž . Ž . Ž . Ž .C %, vrv L cm V ml Q ml min

30.0 200 105 1.430.0 400 135 0.540.0 400 105 1.215.0 300 135 1.020.0 400 105 0.8

( )E. Vereda-Alonso et al.rChemometrics and Intelligent Laboratory Systems 48 1999 81–9084

( )E. Vereda-Alonso et al.rChemometrics and Intelligent Laboratory Systems 48 1999 81–90 85

Fig. 2. Flow chart of the software’s functions. Boxes indicate separate programs; items within a box indicate functions available from thatŽ . Ž . Ž .program’s menu. Arrows indicate calls possible between programs. a Main Menu, b SIMPLEX optimisation, and c robustness test.

Žpoints a blank and four standard solutions contain-y1 .ing 2.5, 5.0, 7.5 and 10.0 mg ml furfural . Six

blank replicates were measured in order to calculatethe limit of detection, and four replicates for eachstandard solution injected were used to calculate therelative standard deviation. Sample consumption wasestimated from the volume of the loop used in the in-jection valve, while reagent consumption was calcu-

Žlated from the flow-rate at the end of the reactor Q,y1 . Ž .ml min multiplied by the analysis t, min and the

Ž .reagent concentration C, % vrv , and divided by100:

QtCConsumption of reagents .

100

The experimental FI parameters optimised were theŽ .concentration of aniline C, % vrv , the reactor

Ž . Ž .length L, cm , the sample loop volume V, ml andŽthe flow-rate as measured at the waste outlet Q, ml

y1 .min . A simplex with five initial vertices was thusestablished. The normalisation conditions and the ini-tial vertices of the simplex were chosen to coincide

with those used in the earlier manual SIMPLEX op-w xtimisation of this method 19 in order to enable cor-

rect comparison of results.

3. Results and discussion

The flow injection system described in Section 2was used to optimise the variables influencing reac-tion development. The four experimental parametersdescribed in Section 2.7 were optimised relative to theresponse function by using different values for the

Ž .quality coefficients s, c, a, r that favoured somequality variables over others. Thus, one can favour

Žsensitivity and precision over cost and throughput as. Žin research laboratories , or vice versa as in indus-

.trial laboratories , or favour sensitivity and through-Žput over precision and cost as is usual in optimising

.flow systems . The same initial values for the param-Ž .eters C, L, V and Q were used in all experiments

Ž .in each optimisation method Table 1 . Since therewere four experimental parameters, five experimentswere needed to form the initial simplex.

( )E. Vereda-Alonso et al.rChemometrics and Intelligent Laboratory Systems 48 1999 81–9086

3.1. Software

The structure of the software, shown in Fig. 2, isbased on a screen menu system, which add to thesystem great flexibility besides simplifying its use.Each routine was called from a menu of up to eightitems displayed on the screen. The integrated systemcan essentially be described by the coupling of threemodules: introducing data for the construction of thecalibration curves and the response functions, max-imisation of the proposed response function by meanSIMPLEX algorithm, and realisation of the rugged-ness test. The system is initiated by the introductioninto it of different pre-established parameters, theR and R values of the seven different vari-max min

ables involved in the response function. Next, the usermust introduce the experimental parameters to opti-mise, both operations are done from Main Menu.

During the optimisation process, the user can modifythe values of a new experimental conditions or to giveto the response function the value of 0.00. This per-mit establish boundaries, given by the user, assigningto the response function the value of 0.00 when theseare overcome. Besides, the program permits savingthe response functions and displaying the graph of the

Ž .response function vs. number of experiments Fig. 3 ,at any time of the optimisation. When the simplex hasfinished, the user can then access to the ruggednesstest. This access is done from the Function menu, thesoftware demands new experimental conditions, ob-tained according to the Youden–Steiner ruggednesstest. This menu permits saving the data of the re-sponse functions at any time of the test. The user can

Ž .see the graph of the ruggedness test Fig. 4 and ob-tain a single optimum value for each experimentalparameter, according to the selection made. In the fi-

Ž .Fig. 3. Variation of the response functions with the number of experiments in the optimisation of the four sets of quality coefficients. a SetŽ . Ž . Ž . Ž .1, b set 2, c set 3, and d set 4. The five response functions optimised for each set are denoted by B .

( )E. Vereda-Alonso et al.rChemometrics and Intelligent Laboratory Systems 48 1999 81–90 87

Ž . Ž . Ž . Ž .Fig. 4. Graphs of the robustness tests for the four sets of quality coefficients optimised by the simplex. a Set 1, b set 2, c set 3, and dset 4. Percentage values represent the contributions of the different parameters to the overall variability. The high and low value of each

.Ž . Ž .parameter are denoted by ( and , respectively.

nal stage, new weights are given to the quality objec-tives of the response function repeating all theabovementioned stages. Thus, the user obtains a set

Ž .of optimised set-ups Table 2 , corresponding to dif-ferent quality compromises.

3.2. Optimisation procedure

Optimisation procedures based on different com-binations of the quality coefficients were carried outin the usual way: the experimental set-up required by

Table 2Experimental variables optimised

y1Ž . Ž . Ž . Ž .C %, vrv L cm V ml Q ml min

Optimum Fixed Optimum Fixed Optimum Fixed Optimum Fixedrange value range value range Value range value

Set 1 22.5–32.0 23.0 212–343 219 75–105 77 1.1–1.6 1.6Set 2 18.8–37.5 19.7 283–400 289 105–135 107 0.5–1.1 1.1Set 3 21.8–39.6 22.7 110–300 120 75–85 76 1.8–2.7 2.7Set 4 20.0–40.0 21.0 200–400 210 101–135 103 0.5–1.4 1.4

( )E. Vereda-Alonso et al.rChemometrics and Intelligent Laboratory Systems 48 1999 81–9088

the optimisation algorithm was assembled, the exper-Žimental responses data for constructing of the cali-

bration curve, relative standard deviation of a stan-dard, sample consumption, reagent consumption and

.time of the analysis were measured, and all valueswere interactively input into the computer program byselecting the input values item from Main Menu inorder to calculate the next experimental conditions.

The following conditions were established asbounds for the optimisation: a maximum analysistime of 4.5 min and a limit of detection not higherthan 10 mg mly1 furfural, which corresponds to theallowed limit in alcoholic drinks; if these values areexceeded, then the user will assign the value 0.00 tothe response function. Based on earlier manual SIM-

w xPLEX optimisation of this method 19 , the normali-sation ranges used were: 0.10–1.17 for the slopes;0.20–10.00 mg mly1 for the limit of detection;1.60–13.22% for the relative standard deviation ofthe slope of the calibration curve; 0.60–7.00% for therelative standard deviation of the analysis; 0.40–1.40ml of pure aniline for reagent consumption; 0–60 mlŽ .plus 75 ml as inner volume of the injection valve forthe sample consumption; and 0.8–4.5 min for theanalysis time.

Four optimisation procedures involving differentŽ .values for the quality coefficients Table 3 were car-

ried out. In set 1, all quality variables were equallyimportant; in set 2, sensitivity and precision were

Žfavoured over cost and throughput as in a research.laboratory ; in set 3, cost and throughput were

Žfavoured over sensitivity and precision as in an in-.dustrial laboratory ; and in set 4, sensitivity and

Žthroughput were favoured over precision and cost as.in the optimisation of a flow system . Fig. 3 shows

the variation of the response function with the num-ber of experiments. It should be noted that, since thereare four variables to be optimised, the simplex hasfive vertices, so any experimental points before the

Table 3Values for the quality coefficients

s c A r

Set 1 100 100 100 100Set 2 100 50 100 50Set 3 50 100 50 100Set 4 100 50 50 100

Table 4Robustness test

Experimental variables Experiment

1 2 3 4 5 6 7 8

Ž .C %, vrv ; C, c C C C C c c c cŽ .L cm ; L, l L L l l L L l lŽ .V ml ; V, Õ V Õ V Õ V Õ V Õ

y1Ž .Q ml min ; Q, q Q Q q q q q Q QResponse function s t u Õ w x y z

sixth do not reflect the ability of the SIMPLEXmethod to climb the surface. The figure does notshow the experiments rejected by the SIMPLEXmethod.

Table 2 shows the optimised ranges for the exper-imental variables in each set of quality coefficients,in addition a special value for each experimentalvariable obtained by shortening the optimized rangeby 90% and asking the user whether a low or a highvalue of the variable was to be used. In the table, alow value was chosen for the aniline concentration,reactor length and sample loop, and a high value for

Žthe flow-rate although all values were good, these.also increased enhance productivity .

3.3. Robustness test

Four robustness tests, one per set of quality coef-ficients, were carried out following SIMPLEX opti-misation. Each robustness test required eight experi-ments since there were four experimental parameters.The experiments are described in Table 4, wherecapital and small letters denote higher and lower val-ues, respectively, optimized by SIMPLEX for eachexperimental parameter. A linear combination of the

Ž .results obtained Table 4 led to the following val-ues:

Cs sq tquqÕ r4 cs wqxqyqz r4Ž . Ž .Ls sq tqwqx r4 ls uqÕqyqz r4Ž . Ž .Vs squqwqy r4 Õs tqÕqxqz r4Ž . Ž .Qs sq tqyqz r4 qs uqÕqwqx r4Ž . Ž .where C, L, V and Q are the experimental FI param-eters described in Section 2.7; and s, t, u, Õ, w, x, yand z are the values of the response functions ob-tained in each of the eight experiments.

( )E. Vereda-Alonso et al.rChemometrics and Intelligent Laboratory Systems 48 1999 81–90 89

Fig. 5. Graphical representation of the optimal ranges for the 2nd and 3rd set of quality coefficients. The individual optimum values for eachparameter are jointed by solid and dotted line, respectively.

The results of the robustness tests are shown inFig. 4; the result of each parameter was divided intothe overall mean in order to normalize all figures.Black circles correspond to the high parameter valueand white circles to the low value. Robustness is re-lated to the distance between points: the shorter thedistance, the greater is the robustness. In these graphs,the contribution of each parameter to the overallvariability is represented by the area inside each rect-angle.

3.4. Interpretation of the data set

Ž .Based on the optimal ranges found Table 2 , thetwo most different situations are represented in Fig.Ž .5 2nd and 3rd set of quality coefficients . The figure

shows that different quality objectives lead to differ-ent experimental set-ups; thus, favoring sensitivityand precision over cost and throughput entails using

Ž .greater reactor lengths better precision and longerŽ .sample loop better sensitivity than if cost and

throughput are to be favored over sensitivity and pre-cision. Also, a lower flow-rate results in poorerthroughput.

Fig. 4 shows that the most robust variable is thesample loop while the least robust ones they arereagent concentration and flow-rate. Also, robustness

varies with quality objectives. Thus, when precisionis favored over throughput, the reactor length is morerobust than the flow-rate; in the opposite situation, theflow-rate is more robust than the reactor length.

4. Conclusions

A number of conclusions can be drawn from theabove-described experiments. First, for thorough op-timization of systems involving four or more vari-ables, the SIMPLEX method may still be the best op-tion; however, the procedure can be high time-con-

Žsuming if done manually computerized optimization.is faster and much less labor-intensive . Second, it is

advantageous to use an interactive optimisation pro-cedure. The proposed integrated software that can beapplied to any analytical method because the user canchoose the experimental parameters to be optimisedand whether a high or low value is to be employedof each parameter; at the end of the process, com-puter provides a single optimum value for each pa-rameter and additional information about its robust-ness. In addition, system can be adapted to specificquality requirements by the user simply by changingthe quality coefficients from the Main Menu. The

( )E. Vereda-Alonso et al.rChemometrics and Intelligent Laboratory Systems 48 1999 81–9090

user can also change the convergence criterion to beused in the simplex process. Third, the system allowsthe response function to be selected; even though all

Žthe quality variables dealt with here sensitivity, pre-.cision, cost and sample throughput are well-known,

they have rarely been considered as a whole or in asystematic manner. The integrated use of mathemati-cal modeling and optimisation algorithms allows dif-ferent optimal configurations for a given analyticalsystem to be obtained, if sufficient response func-tions are generated. However, the best option de-pends on the quality preferences set by the user andon efficient adaptation of the set-up to each casestudy; in any case, the final selection will be an opti-mal set-up as regards its quality objectives. The abil-ity to select the most appropriate set-up for each casestudy highlights the potential of the proposedmethodology. Fourth, robustness tests determine thesignificance of optimised experimental variables tochanges in the response of each analytical system,which provides the user with additional informationregarding which parameters require specially carefulcontrol.

Acknowledgements

ŽFinancial support provided by the DGICyT Pro-.ject No. PB95-0977 is gratefully acknowledged.

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