n Tutorial 91

Chemometrics and Intelligent Laboratory Systems, 8 (1990) 97-107 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

Optimization Using the Super-Modif ied Simplex Method

EDWARD MORGAN * and KENNETH W. BURTON

Department of Science and Chemical Engineering, The Polytechnic of Wales, Trejorest, Mid-Glam, CF37 IDL (U.K.)

GRAHAM NICKLESS

Department of Inorganic Chemistry, School of Chemistry, The University, Bristol BS8 ITS (U.K.)

(Received 11 January 1990; accepted 27 February 1990)

CONTENTS

Abstract ............................................ 1 Introduction ........................................

1.1 Review of the modified or variable-size simplex procedure ..... 2 The super-modified simplex procedure .....................

2.1 Rules for movement of the super-modified simplex .......... 2.2 Boundary violations by the super-modified simplex ...........

3 Worked example ..................................... 4 Conclusions ........................................ References.. .........................................

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ABSTRACT

Morgan, E., Burton, K.W. and Nickless, G., 1990. Optimization using the super-modified simplex method. Chemomet- rics and Intelligent L.aboratory Systems, 8: 97-107.

The super-modified simplex method represents an advance over the modified simplex method. It has an ability to change its size and orientate itself to fit a response surface by second-order and gaussian estimation of the position of

an optimal vertex from previously obtained responses. The super-modified simplex is essentially a set of rules which,

like other simplex methods, can be presented as a flow chart. Restrictions to the positions that the optimal vertex may

take are also presented since these enable the super-modified simplex to maintain its symmetry during the optimization process and make effective progress towards an optimum. Attempted boundary violations by both the second-order

and gaussian super-modified simplex methods are also dealt with since they behave quite differently to other simplex

techniques. Finally an example is presented with some of the required calculations to demonstrate movement of the

simplex across a response surface. This tutorial should act as an introduction to the super-modified simplex method for

students and scientists who are interested in learning about the technique without having to search through all the

available literature.

0169-7439/90/$03.50 0 1990 - Elsevier Science Publishers B.V.

98 Chemometrics and Intelligent Laboratory Systems m

1 INTRODUCTION

Optimization of operating conditions for the parameters or variables affecting many chemical

systems occupies a large proportion of the time and effort of many chemists. Many of these chem- ical systems are affected by more than one factor, and the chemist attempts to search for a combina- tion of these which optimizes the system or leads to the best response. Several exploratory tech- niques are available to chemists including steepest ascent methods, incorporating factorial designs and/or specific response surface designs, such as Box-Wilson designs [l-4]. However, these designs usually require many experiments when the num- ber of variables to be optimized is large. In con- trast to this a host of sequential simplex methods have been developed over the last thirty years, based upon a technique originally developed by Spendley and co-workers [S]. These simplex meth- ods generally require less experiments to find an optimum, especially when the number of variables to be investigated is large. In previous tutorial

articles the basic or fixed-size simplex and mod- ified or variable-size simplex procedures have been described [6,7]. Readers are directed to these if they are totally unfamiliar with the simplex method. In addition there are also several reviews which have covered the modified simplex proce- dure [8-lo].

nl

b I2 d

Factor 1 Fig. 1. Modified simplex for two factors showing reflection to vertex R, expansion to vertex E, contraction towards the reflec- tion at vertex C, and contraction towards the worst vertex at

Cw.

The original basic simplex technique is fairly limited in its applicability to most chemical sys- tems, with its main drawbacks being an inability to adapt quickly (in few experiments) to the shape

of the response surface and an inability to accel-

erate across the response surface when responses indicate that this is favourable. Recognizing these problems Nelder and Mead developed the mod- ified or variable-size simplex procedure [ll]. This was further modified in attempts at improving its behaviour on a variety of response surfaces [12]. As a result it has now been widely applied in a variety of situations [13-151. The modified sim- plex technique forms the foundation on which the super-modified simplex procedure is based and will be briefly described here.

1. I Review of the modified or variable-size simplex procedure

Fig. 1 shows the moves that a two-factor sim- plex may make according to the response obtained at its reflected vertex. If the response at vertex R

is more desirable than the response at B, the best response in the current simplex, an expansion to E is attempted. On the other hand, if the response at R is less than that obtained at B, contraction of the simplex is attempted on the assumption that the simplex has moved too far in one direction, perhaps over-spanning the optimum or moving off the edge of a ridge on the response surface. In the modified simplex two contractions are possible, depending on the response at R relative to those at N and W, either towards the worst vertex at C, or towards the reflection vertex at C,. In fact, if

the response at R is less than at B, but not less than at N, no expansion or contraction is at- tempted and the reflection vertex replaces the worst vertex in a new simplex figure. Similarly, where expansion to vertex E does not lead to an improvement in the response, it is said to have failed and the reflected vertex used in the next simplex. Failed contractions are dealt with by rejecting the next-to-worst vertex in a simplex figure containing the contracted vertex.

There is also a rule which states that if a vertex is retained in k + 1 simplexes, where k is the number of variables, it is re-tested and, depending

W Tutorial 99

on the response obtained, is either retained in or rejected from the simplex. This rule has its origins in the basic simplex procedure and is used when it is likely that the responses obtained are subject to error or noise. A vertex may then have an inflated response and be retained in the simplex when it should have been rejected. A possible consequence of this is that the simplex can circle around a false optimum. These rules have been expressed as a flow chart in a previous tutorial article by the authors [7].

Another rule in the modified simplex procedure deals with boundary violations by the reflected and expansion vertices in which one or more of the calculated coordinates for a vertex is either lower or higher than is practically possible for that variable; the simplex is attempting to move across one of the boundaries. In such a case an experi- ment is ‘not carried out and a low or negative response is assigned instead. This will cause the simplex to contract towards the worst vertex so that a new vertex will be contained within the original simplex and therefore within the boundaries of all the variables.

2 THE SUPER-MODIFIED SIMPLEX PROCEDURE

2.1 Rules for movement of the super-modified sim- plex

After an experiment has been carried out at the reflection vertex in the modified simplex proce- dure an experiment is carried out at one of three possible vertices, at an expanded vertex E, at a contraction vertex towards the reflection C,, or at a contraction vertex towards the worst vertex C,. These are all carried out on a line from the worst vertex W, through the centroid conditions P, to the reflection vertex R. The positions of the coor- dinates of the centroid may be calculated from the following equation:

where j is a variable which varies from 1 to k, i is

(1)

a vertex which varies from 1 to k + 1, p is the centroid and w is the vertex with the worst re- sponse. It is also possible to represent the vertices by vectors of the coordinates. For a simple two- factor case, the centroid conditions are repre- sented by a vector obtained by averaging the responses remaining once the worst vertex has been discarded such that:

P(B + N)/2 (2)

The coordinates of any of these vertices may then be calculated using the following equation:

V better, j = Y*v,*j+(l- Y)*vW.j (3)

where ‘better j , refers to the value for the jth variable of the reflected vertex, Y is the expansion factor referred to in Fig. 1 which takes different values according to the type of vertex required: Vertex R: Y = 2;. Vertex E: Y = 3; Vertex C n: Y = 1.5; Vertex C w: Y=O.5. In vector notation the coordinates of any of these vertices (Y) are given by:

Y=Y*P+(l-Y)*W (4

The super-modified simplex as first described by Routh et al. [16] takes this further and uses an optimal value of Y, from here on referred to as Y opt, instead of only one of the three possibilities. Using a Y,,, value should enable the simplex to follow the response surface more closely than the modified simplex procedure. To calculate Y,,, it is necessary to carry out an additional experiment at the centroid conditions, so that there are three responses obtained on line Y. From three re- sponses it is possible to fit a second-order poly- nomial to find a position on line Y which maxi- mizes the response (Fig. 2). The experimenter now carries out an experiment at the coordinates dic- tated by this Yopt value. The coordinates for this new vertex (0) may then be found using an adapted eq. (3) such that:

V better. j = yOpt*vp,j+(l- yOpt)*‘w,j

or in vector notation:

0 = Yopt *P+ (1 - Y,,,) *w

(5)

(6)

100 Chemometrics and Intelligent Laboratory Systems n

Y- Fig. 2. Second-order fit of Yopt in super-modified simplex; w is the response at the worst vertex, p is the response at the centroid and r is the response at the reflected vertex.

in which Y,,, is determined from

Y,,, = [(w - P)/(w - 2p + r)] + 0.5 (7)

where w, @ and r are the responses obtained at the worst vertex, centroid conditions and reflected vertex respectively. From eq. (7) it is possible that Y,,, may represent a value of Y which does not maximize the response, as shown in Fig. 3. To avoid this the condition w-2p+ r < 0 should hold.

Using a potentially infinite number of values

for Y&t should enable the simplex to follow the response surface more closely than the three posi- tions available in the modified simplex procedure. However, when no restrictions are placed upon the minimum and maximum values that YoPt can take it is possible for a minor change in a response to have an unduly large effect on the simplex movement. To avoid this it is usual to set a minimum of -1 and a maximum of 3 on the value of YoPt. These values allow the simplex to grow until it becomes twice the size of the simplex from which it originates [17].

When the value of Y,,, is close to 1, the new optimal vertex (0) will be located near the centroid p. If this optimal vertex has a better response than

,w b-L-4 /w !P r /

Y- Fig. 3. Arrangement of responses for which a maximum cannot be found in a second-order fit.

1 -1 0

/ 1 2 3

Expansionfactor Y +

Fig. 4. Restrictions to the Y factor in a super-modified simplex. The shaded areas are restricted.

the reflected vertex it will be retained in the sim- plex and may cause the simplex to lose the ability to move in a certain direction. In such a case the simplex is said to degenerate. For example, a two-dimensional simplex, a triangle, will effec- tively become a straight line joining the previously best vertex and next-to-worst vertex. This should be especially avoided in the early stages of optimi- zation. Similarly, when a value close to zero is calculated for YoPt , the optimal vertex will be located near the vertex with the worst response and progress will be made by the simplex. To avoid both these situations safety margins around Y = 1 and Y = 0 have been introduced [17]. The safety margin is always less than 0.5 and typically 0.1 or 0.3. Fig. 4 shows the safety margins which are typically applied to a Yopt after it has been calculated initially. For example, if a Yap, value of 0.85 is calculated, applying a safety margin of 0.3 will cause the Y,,, value to be adjusted to 0.7. These restrictions should allow the simplex to maintain a degree of symmetry during the optimi- zation process. It is possible to remove the centroid restriction on the Y,,, value in the later stages of optimization, when the problem of degeneracy is less important, according to the criteria which are used to terminate the process.

The super-modified simplex procedure as de- scribed to this point represents the procedure which is generally accepted as standard and a flow chart is presented in Fig. 5. However, many re- searchers have suggested other improvements, in- cluding estimation of the response at the centroid (p> by the average of the k responses remaining when the worst vertex is discarded [18]. This avoids carrying out the experiment at the centroid condi- tions which is the only real disadvantage of the super-modified simplex procedure. The efficiency

n Tutorial 101

c-3 START

Rank b,n&w -

Re-test

%%F

ish

3 itrOES inorder

I I ’

Evaluate Yoptand adjustfor constraints

Fig. 5. Flow chart of super-modified simplex algorithm. Capital letters denote the vertex proper and lower-case letters the

response; “ >” should be interpreted as “better than”. If an

answer is “Yes”, the path in the flow chart is downwards; if an answer is “No”, the path is to the side.

Y-

Fig. 6. Gaussian fit of YOP, in super-modified simplex.

of this procedure compared favourably with the standard super-modified simplex in that it con- verged on the optimum more often in less experi- ments over a wide range of starting conditions for both theoretical response surfaces and practical chemical systems 1181.

Another variation of the super-modified sim- plex, which may suit some response surfaces, uses a gaussian estimate of Yopt in place of a second- order estimate [19]. A situation where a gaussian estimate of YoP, may estimate the optimum posi- tion better than a second-order estimate is given in Fig. 6. The equation to calculate the gaussian Yap, value is given by:

YoP, = [(In w - In p)/(ln w - 2 In 3 + In r)] + 0.5

(8)

To find a value of Yopt which maximizes the response, the condition In w - 2 In p + r c 0 must hold true. As with the second-order estimate re- strictions are placed upon a gaussian YO,,( value to avoid the simplex becoming too large, degener- ating or making no progress. Comparison of second-order and gaussian Yopt values calculated

with various responses at W, p and R has shown that gaussian Y,,, values have greater working ranges than second-order YoP, values [19]. When the two procedures were compared on several theoretical response surfaces it was found that the gaussian super-modified simplex was more effi- cient at finding the optimum [19].

102 Chemometrics and Intelligent Laboratory Systems m

2.2 Boundary violations by the super-modified sim-

plex

When the simplex attempts to cross a boundary in the modified simplex procedure the vertex is assigned a low or negative response. It is, however, possible that the optimum response coincides closely with the boundary constraint of one or more of the variables. Also, assigning a low or negative response is not in itself a cure to the problem since movement of the simplex is dictated by these responses. To get around this problem the super-modified simplex deals with the crossing of a boundary in a completely different way. First of all the Y factor corresponding to the crossed boundary is calculated from

where Y, is the boundary Y factor and B, is the boundary value for variable j. When the super- modified simplex was originally introduced by Routh et al. [16] this Y, value was accepted as the value to be used in the calculation of the condi- tions for both boundary-crossing reflection and optimal vertices. A constrained reflection experi- ments could therefore be carried out at the boundary conditions indicated by the constrained Y value in Fig. 7. It has since been suggested that Y, should only be accepted for the calculation of reflection conditions if it has a value greater than or equal to 1.5. If Y, < 1.5 then a Y value of 0.5 is used instead. This restriction is used to ensure a reasonable spacing between R, p and W [17].

I Ya,

I I 0 1 2

------,Y

Fig. 7. Boundary violation of a reflected vertex in a two-factor super-modified simplex.

However, this is not the end of the problem as far as constrained reflection vertices are concerned since the second-order or gaussian estimate of YoPt which is used to calculate the position of the optimal vertex depends upon an even spacing of R, p and W. When a constrained reflected vertex has been encountered the equations for calculating Y,,, have to be adjusted to take this into account.

For second-order Yap, estimates, YoPt is now calculated from:

Yap, = 0.5( Y; - YJ * (w - p)

/[(Y,-l)*w- Y,*p+r] +0.5 (IO)

with the additional condition that when Y, > 1, p

must be greater than [r + (Y, - 1) *w]/ Y, or, if Y, is less than or equal to 1 and Y, does not equal zero, p must be less than [r + (Y, - 1) *w]/ Y,. If either of these conditions is not met a maximum value of Yopt cannot be calculated and is instead set to either -1 or 3 [17].

For gaussian YoP, estimates in which a reflected vertex has attempted to cross a boundary, Y,,, is now calculated from:

Y0,,=0.5(Yi- Y,)*(lnw-lnp)

/[(Y,-l)* lnw- Y,* lnp+lnr] +0.5

(11)

with the additional condition that if Y, > 1, 3 must be greater than r(‘/y~ *w(‘-~/~~), or if Y, is less than or equal to 1 and Y, does not equal zero, p must be less than r”/yo) *w(‘-~/~). If either of

these conditions is not met a maximum value of

Y,,, cannot be calculated and is instead set to either - 1 or 3 [17]. If Y, is set to 2, as it would be if a boundary had not been crossed, eq. (10) and (11) simplify to those given previously for calculat- ing the second-order and gaussian estimates of YoP, respectively.

If an expanded vertex at Y = 3 is indicated in the modified simplex procedure and this lies out- side a boundary for one of the variables, a low or negative response is also assigned to E (Fig. 1). The reflection vertex inside the boundary is then retained. However, when a Y,,, value of greater than 2 is indicated in the super-modified simplex

procedure’and Y,,, lies beyond the boundary, Y,,,

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TABLE 1

Worked example: initial simplex coordinates and responses

Vertex Rank AF H Absorbance

1 W 5.000 1.000 0.190 2 N 6.000 1.000 0.210 3 B 5.500 1.866 0.220

4 P 5.750 1.433 0.215 5 R 6.500 1.866 0.245 6 0 7.250 2.299 0.270

Comment

Maximum not possible, Yap, = 3 o > therefore simplex r, BNO

may be assigned to the boundary conditions (Y,), calculated from eq. (9). The experimenter then

carries out an experiment at these boundary con- ditions (Fig. 8) and the procedure continues as before [17].

3 WORKED EXAMPLE

Having introduced the reader to the super-mod- ified simplex method, a worked example will now be presented in which the reader should be able to follow some of the decisions used to move the simplex around a response surface, working out

the required calculations at each step, and finally how the procedure is terminated.

In the optimization of a flame atomic absorp- tion spectrophotometer (AAS) for chromium de- terminations in a set of samples from a complex organic matrix, two parameters were identified as being important. These were the air to fuel ratio (AF) and the height of the burner in relation to the beam of radiation emanating from the hollow cathode lamp (H). The response used in this study was the absorbance value read-out from the AAS on the basis of a 51.18 ml-’ standard solution being aspirated. As a large number of samples were to

be analysed, the only method available to the analyst was to use matrix-matched standards. Thus

the 5pg ml-’ solution contained some of the background materials to be expected in the dis- solved sample solutions.

A previous best response had been obtained at an air-to-fuel ratio of 5.000 and a burner height of 1.000 cm. Both these parameters could be set electronically on the instrument. The next step for the experimenter was to select suitable step-sizes

which could give reasonable changes in the re- sponse. These were chosen to be 1.000 for both AF and H.

A lower boundary constraint for the air-to-fuel ratio was identified around 4.000, at which point the flame becomes sooty (and potentially explo- sive!). The boundaries for burner height were 0.000 cm, where radiation from the hollow cathode lamp is incident upon the burner, and 10.000 cm, the greatest movement possible. The worked example used here is the same as that presented in the tutorial on the modified simplex method except that both variables are now quoted to three deci- mal places.

Coordinates of the initial simplex were calcu- lated as previously by reference to the table of Long [20]. Since the initial conditions for AF and H were 5.000 and 1.000 respectively, with both having step-sizes of 1.000, this gave the following experimental conditions.

Vertex AF H

1 5.OOc+o = 5.000 1.000+0 =l.OOO 2 5.000+1*1.000 =6.000 l.OOC+O =l.OOO 3 5.000+0.5*1.000 =5.500 1.000+1*0.866 =1.866

The experimental conditions defined by this initial simplex were run, the absorbances obtained were noted and are shown in Table 1. The response of 0.190 for vertex 1 agreed well with what had been previously achieved at these conditions. The re- sponses to vertices 2 and 3 were encouraging but did not represent the substantial improvement in the response which the experimenter had hoped for. The next stage was to rank the vertices as Best, Next-to worst and Worst according to the responses obtained (Table 1). The coordinates of P were defined earlier as the average of the sum of

104 Chemometrics and Intelligent Laboratory Systems n

/__. ._-_T__ Ya, Yopt ,

0 1 2

---a y

Fig. 8. Boundary violation of an optimal vertex in a two-factor super-modified simplex.

the remaining vertices having removed the worst vertex such that:

N= (6.000 1.000)

B= (5.500 1.866)

N+B= (11.500 2.866)

Therefore, P = (5.750 1.433) An experiment was conducted at these coordinates and an absorbance of 0.215 obtained (Table 1). The coordinates of the reflected vertex were then obtained from eq. (4).

R = 2**+ (1 - 2)*W = (6.500 1.866)

A new experimental run defined by this reflected vertex was conducted and an absorbance of 0.245 recorded (Table 1). It should now have been possi- ble to fit a second-order polynomial to the three responses to find Y,,, but the condition w - 2$ + r -C 0 had first to hold true, otherwise a maximum could not be calculated. Since

w - 2@ + r = 0.19 - (2*0.215) + 0.245 = 0.005

the condition did not hold and a value of Yopt which maximized the absorbance could not be

TABLE 2

Worked example: second simplex coordinates and responses

found. A value of Y,,, therefore had to be as- signed for use in eq. (5) or (6). The response at the reflected conditions was greater than at both the previous worse and centroid conditions, indicating that an expansion of the simplex beyond the re- flected conditions might have led to an improved response. Therefore, a Y,,, value of 3 was used in eq. (6) so that

0=3*P+(1-3)*w

The coordinate for the air-to-fuel ratio variable (AF) in the optimal vertex was therefore:

0,F=3*5.750+(-2)*5.000=7.250

and for burner height (H):

On=3*1.433+(-2)*1.000=2.299

so that the optimal vector was given by

0 = (7.250 2.299)

This experiment was run and an absorbance of 0.270 obtained (Table 1). The response was com- pared with that of the reflection vertex and since o was greater than r, vertex 0 was retained in a new simplex containing vertices BNO (see Fig. 5).

The criterion used to stop the simplex optimi- zation was the coefficient of variation (COV), standard deviation of the responses in the simplex expressed as a percentage of the mean response. To facilitate comparison with the same example optimized via the modified simplex procedure it was decided to have the same optimization crite- rion, that is to halt the simplex when the COV became less than 0.5%. For the second simplex defined by BNO (vertices 2, 3 and 6), the COV value was 13.8% (Table 2). The procedure could therefore continue.

The second simplex was then considered and the process restarted by ranking the vertices

Vertex Rank AF H Absorbance Comment

2 W 6.000 1 .oOfl 0.210 3 N 5.500 1.866 0.220 6 B 7.250 2.299 0.270 7 P 6.375 2.083 0.250 8 R 6.75 3.165 0.302 9 0 7.125 4.248 0.360

COV value = 13.8%

Maximum not possible, Y,,, = 3 o > r, therefore simplex BNO

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4 5 6 7 6 9 IO

A/UFue/Rafb ( AF ) Fig. 9. Worked example: progress of the super-modified sim- plex across the response surface for two factors (AF= air-to- fuel ratio, H = burner height) with contours of absorbance values.

according to their responses as B, N and W (Table 2). The calculations necessary for this second sim- plex were much the same for the initial simplex.

Again no maximization of the response was possi- ble and a YoP, value of 3 had to assigned. The experiments were carried out, with an optimal vertex (vertex 9) eventually being retained (Table 2). At this point a substantial improvement in the absorbance over the initial absorbance had al- ready been obtained.

Fig. 9 is a response surface diagram which has absorbance value contours as functions of the values of the two factors AF and H. It is possible to follow the movement of the super-modified

TABLE 3

Worked example: third simplex coordinates and responses

simplex across this response surface by examining the numbers of the vertices. No vertex is shown unless it has been retained. For example, the re- flection vertex of the initial simplex (vertex 5) is not shown, only the optimal vertex (vertex 6), which was retained in its place.

In the third simplex a Y,,, value of 1.09 was calculated from the second-order polynomial. An optimal vertex experiment carried out at these coordinates, close to the centroid, might have led to it being retained instead of the reflected vertex (if it had a greater absorbance). This could have limited the number of dimensions the simplex could effectively move in. Therefore, a safety margin of 0.3 was applied and a Y,,, value of 1.3 used instead (Table 3).

As can be seen in Fig. 9 the simplex procedure continued until the simplex successfully homed in on a relatively small area of maximum absorbance (0.400). The coordinates of the vertices and re- sponses of the fourth simplex up to and including the final simplex (simplex 9) are given in Table 4. The reader should now be able to follow the

calculations and the movement of the simplex to this final area by reference to both table 4 and Fig. 9 at which point the termination criterion had

been met (all vertices had the same absorbance of 0.400), indicating that no improvement was possi- ble.

The optimization was fairly rapid in that only 27 experiments were required before all the vertices lay in the optimal region of absorbance. This was slightly more than required by the modified sim- plex starting from the same position [7], although rather less simplices were required than for the modified simplex. This is not really that surprising since this example appears to take place on a

Vertex Rank AF H Absorbance

3 W 5.500 1.866 0.220 6 N 7.250 2.299 0.270 9 B 7.125 4.248 0.360

10 P 7.188 3.273 0.320 11 R 8.875 4.681 0.250 12 0 7.694 3.695 0.340

Comment

COV value = 25.0%

Y,,, = 1.09, make YoP, = 1.3 o > r, therefore simplex BNO

106 Chemometrics and Intelligent Laboratory Systems R

TABLE 4

Worked example: coordinates and responses for simplices 4-9

Vertex Rank AF H Absorbance Comment

Simplex 4

Simplex 5

Simplex 6

Simplex 7

Simplex 8

Simplex 9

6 9

12

13 14 15

7.250 2.299 0.270 7.125 4.248 0.360 7.694 3.695 0.340

7.409 3.971 0.350 7.569 5.644 0.330

3.457 4.473 0.390

9 12 15

16 17 18

7.125 4.248 0.360 7.694 3.695 0.340

3.457 4.473 0.390

7.291 4.360 0.373 6.888 5.025 0.360 7.170 4.560 0.375

9 15 18

19 20 21

15 21 23

25 26 27

W B N

P R 0

N W B

P R 0

W B N

P R 0

N W B

ii R 0

W B N

P R 0

7.125 4.248 0.360 3.457 4.473 0.390 7.170 4.560 0.375

7.314 4.517 0.383 7.502 4.786 0.395 7.614 4.944 0.400

15 18 21

22 23 24

3.457 4.473 0.390 7.170 4.560 0.375 7.614 4.944 0.400

7.314 4.517 0.383 7.901 4.858 0.400 7.718 4.783 0.400

3.457 4.473

7.614 4.944 7.901 4.858

7.757 4.901 8.059 5.329 7.657 4.759

21 7.614 4.944 23 7.901 4.858 27 7.657 4.759

0.390 0.400 0.400

0.400 0.350 0.400

0.400 0.400 0.400

COV value = 14.6%

Yap* = 1.3 o > r, therefore simplex BNO

COV value = 6.93%

Y,,, = 1.217, make YO,,, = 1.3 o > r, therefore simplex BNO

COV value = 4.00%

Yopt = 2.591 o 1 r, therefore simplex BNO

COV value = 3.24%

Yopt = 1.5 o = r therefore simplex BNR

COV value = 1.45 %

Y,,, = 0.67 o > r, therefore simplex BNO

COV value = 0.0%

rather simple response surface and was used prim- arily to demonstrate the calculations involved. Only on a more complex response surface which has rapid changes in the slopes of the response surface and in the directions of the optimum path might the super-modified simplex show its ad- vantages over the modified simplex in being able to accelerate and decelerate to follow the response surface.

4 CONCLUSIONS

In this tutorial the modified simplex method was briefly reviewed as a basis for the super-mod- ified simplex. The concept of an optimal vertex was introduced to replace the rather limited op- tions available in the modified simplex. Restric- tions to the actual position that this optimal vertex may take in the super-modified simplex were de-

n Tutorial 107

scribed since these have been shown to lead to an improved performance. The gaussian method of obtaining a Y,,, value was also described since this can be useful on some types of response surface. Finally a worked example was presented with some example calculations of second-order YOPt values. The reader should be able to follow these calculations and follow the progress of the super-modified simplex across a response surface.

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G.E.P. Box and N.R. Draper, Evolution Operation, A Stat-

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G.E.P. Box and K.B. Wilson, On the experimental attain-

ment of optimum conditions, Journal of the Royal Statisti- cal Society, Series B, 13 (1951) l-45. D.L. Massart, A. Dijkstra and L. Kaufman, Evaluation and

Optimization of Laboratory Methoak and Analytical Proce-

dures, Elsevier, Amsterdam, 1978.

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