Manual
for the Simplex Method
Tadeusz Michałowski *, Maciej Rymanowski and Andrzej Pietrzyk
Notations: xij – number (scalar); xi – vector, X - matrix
1. Introductory to the Simplex Method
1.1. Some fundamental concepts and data known from the geometry course
In order to introduce the simplex concept, we start from the point (0-D), (I) segment (1-D), (II) equilateral triangle (2-D), and (III) tetrahedron (3-D) concepts, known from the elementary geometry.
The 0-D, 1-D, 2-D, 3-D (and generally n-D) notations express the dimensionality of the corresponding
geometrical entities (Fig.M1).
A2
I
A0
A3
A1
II
III
A2
A0
A1
A0
A1
Fig.M1. Segment, equilateral triangle and tetrahedron (tetragon)
The points A0, A1, A2 are vertices of the triangle A0A1A2; the points A0, A1, A2, A3 are the vertices of the
tetrahedron A0A1A2A3.
A0A1 is one of the edges of the triangle; A0A1 is one of the edges of the tetrahedron;
Equilateral triangle A0A1A2 is one of walls of the tetrahedron.
Segment, equilateral triangle, and tetrahedron can be presented in the same scale; it means that all
distances, equal to unity (chosen arbitrarily), between the (neighbouring) points Ai and Aj of a polygon
are assumed, i.e. the length AiAj = 1.
All angles between edges of the equilateral triangle are equal to 60o (π/3 rd). All angles between adjacent tetrahedron edges are equal 60o too, owing to the fact that all the tetrahedron walls are composed of equilateral triangles (2-D). By turns, tetrahedrons form the 3-D–walls in 4-D polygon, etc. It
means that all angles between adjacent edges of a symmetrical n-D polygon are equal 60o, as well.
*
Corresponding author, E–mail address: [email protected]
The equilateral triangle can assume different positions on the plane. A particular case is the triangle
with unit edges, ‘anchored’ at the origin of co-ordinate axis, as one presented in Fig. M2.
The successive vertices of the equilateral triangle can be presented in matrix form, where the rows are
expressed by co-ordinates of the corresponding points Ai(i=0,1,2):
X=
0
1
1/2
0
0
3 /2
A0
A1
A2
(M1)
X2
A2
A0
A1
X1
Fig.M2. The equilateral triangle with unit edges and A0(0,0); A1(1,0), A2(0.5,
3 /2)
The second and third rows of this matrix are identical with components of transposed vectors: A0A1
and A0A2, respectively:
A0A1 = [ 1, 0] T, A0A2 = [ 1/2,
T
3 /2]
The tetrahedron can assume different positions in 3-D space. A particular case is the tetrahedron anchored at the origin of co-ordinate axis as one presented in Fig.M3.
X3
A3
A0
A1
X1
A2
X2
Fig.M3. The tetrahedron with unit edges and A0(0, 0, 0); A1(1, 0, 0), A2(1/2,
A3(1/2,
3 /6,
3 /2, 0);
6 /3)
The successive vertices of the tetrahedron can be presented in matrix form:
2
X=
0
1
1/2
1/2
0
0
3 /2
0
0
0
A0
A1
A2
3 /6
6 /3
A3
(M2)
The second, third and fourth rows of this matrix are identical with components of the transposed vectors: A0A1, A0A2 and A0A3, respectively:
A0A1 = [ 1, 0, 0 ] T, A0A2 = [ 1/2,
T
3 /2, 0 ] , A0A3 = [ 1/2,
3 /6,
T
6 /3 ]
1.2. A generalised approach to the polygon concept in n-D space
The segment, equilateral triangle and tetrahedron are imaginable concepts. However, application of
the concepts involved with geometric figures of higher dimension (n≥4) is beyond imagination. One
should be taken into account that, in the simplex optimisation, we are forced, as a rule, to consider
higher number of factors influencing the optimising process.
Although the figures in n-D space are beyond imagination, one can present them in projective
space. In Fig.M4, any pair of points of the simplex is connected, informally, by a segment (line); the
segments (sides, edges) are marked there with broken lines. Particularly, two points form the 1-D simplex in 1-D space. Adding a third point (that not lies along the segment or its extension) gives a triangle (2-D simplex) in 2-D space. Adding a fourth point in 3-D space (not coplanar with 2-D simplex
plane), one obtains a tetrahedron (3-D simplex), etc. Generally, adding a successive k+1–th
(k=2,3,...,n-1) point, do not lying in surface formed by k–1-D simplex, one obtains the k-D simplex. In
other words, adding the successive point that not lies on (or in plane of) simplexes of lower dimension
and connecting it, every time, with the remaining points, one can imagine the simplexes with larger
and larger dimensions, in projective space (Fig.M4).
3
6
5
1
2
4
Fig.M4. Formation of consecutive dimensions (up to 6-D) of a simplex in projective space
Let us assume that the polygon is ‘anchored’ in n-D–space in such a manner that A0 is in the origin
of the co–ordinate axis and any successive point ‘enters’ the new dimension of the n-D–space, compare with Fig.M3. It means that the co–ordinates of the corresponding points of n-D polygon are as
follows:
3
A0 = (0, 0, 0, ... , 0, 0); A1 = (x11, 0, 0, ... , 0, 0); A2 = (x21, x22, 0, ... , 0, 0); A3 = (x31, x32, x33, ... , 0, 0);
... ; An–1 = (xn–1,1, xn–1,2, xn–1,3, ... , xn–1,n–1, 0); An = (xn1, xn2, xn3, ... , xn,n–1, xnn).
(M3)
This means that xij = 0 for j > i.
Note that the number of vertices exceeds, by 1, the dimension of the space; it means that the polygon in n-D space involves n+1 vertices.
On the basis of the points Ai (I = 0,1,2, …, n) one can construct n vectors xi = A0Ai, originating at A0
and ending at the points Ai . One should notice that
The basic components xij of the vector xi = A0Ai = [xi1, xi2, xi3,…, xii, …, xin]T , starting at the
origin of co-ordinates, are identical with co-ordinates of the point Ai = (xi1, xi2, xi3,…, xij,…, xin).
Then for the vectors A0Ai based on the points specified in (M3) we have:
x1 = A0A1 = [ x11, 0, 0, …, 0 ] T, x2 = A0A2 = [x21, x22, 0, …, 0]T, x3 = A0A3 = [x31, x32, x33, 0, … , 0]T, …
xi = A0Ai = [x31, x32, x33, … , xii,, 0, …, 0]T , … , xn = A0An = [xn1, xn2, xn3, … , xni, …, xnn]T
(M4)
Let us construct the matrix composed of the co-ordinates of the corresponding points in n-D space, as
assumed above:
0
0
0 … 0 … 0
0
x11
0
0 … 0 … 0
0
x21
x22
0 … 0 … 0
0
x31
x32
x33 … 0 … 0
0
……………………………………………….
xi1
xi2
xi3 … xii … 0
0
……………………………………………….
xn-1,1
xn-1,2
xn-1,3 …xn-1,i …xn-1,n-1 0
xn1
xn2
xn3 … xni … xn,n-1
xnn
X=
A0
A1
A2
A3
…
Ai
…
An-1
An
(M5)
On the basis of the above remark, the terms xij in the 2nd, 3rd, …, (n+1)th rows of the matrix X (eq.
M3) are identical with components of the corresponding transposed vectors xi = A0Ai.
1.3. The simplex concept
1.3.1. Scalar product of vectors
Let ek be the unit vector along k–th axis, Xk, i.e. ej=1. The ei vectors form a set of mutually orthogonal vectors in n-D space (Cartesian co-ordinate system). Scalar products:
ei⋅ ej =ei⋅ej⋅cos(∠ei,ej) = 0 for j ≠ i and ei⋅ ei =ei⋅ei⋅cos(∠ei,ei) = 1 for j = i
(M6)
Any vector xi can be also written as follows:
•
in 2–D space:
xi = [xi1, xi2]T or xi = e1⋅xi1 + e2⋅xi2 ;
•
in 3–D space:
xi = [xi1, xi2, xi3]T or xi = e1⋅xi1 + e2⋅xi2 + e3⋅xi3
•
(M7)
(M8)
in n–D space:
xi = [xi1, xi2, ... , xin]T ; xi = e1⋅xi1 + e2⋅xi2 + ... + ek⋅xik + … + en⋅xin
(M9)
4
Scalar product of two vectors, xi and xj, is as follows:
x iT x j
= (e1⋅xi1 + e2⋅xi2 + ... + ek⋅xik + … + en⋅xin)(e1⋅xj1 + e2⋅xj2 + ... + ek⋅xjk + … + en⋅xjn)
= xi1⋅xj1 + xi2⋅xj2 + ... + xik⋅xjk + ... + xin⋅xjn
(M10)
(see eqs. M6). Particularly, for j = i
x iT x i
= xi12 + xi22 + ... + xik2 … + xin2
(M11)
If the unit vectors xi are assumed, i.e.xi = 1, then
xi12 + xi22 + ... + xik2 + … + xin2 = 1
(M12)
If the unit vectors xi as ones specified in eq. M4 are applied, then from eq. M11 we have:
xi12 + xi22 + ... + xii2 = 1 (for i = 1,…,n)
(M13)
The scalar product of vectors xi and xj can also be written as follows
xi⋅xj =xi⋅xj⋅cos(∠xi,xj)
(M14)
where ∠xi,xj is the angle between vectors xi and xj. For the unit vectors we have:xi = xj = 1.
As was stated above, the angle between vectors xi = A0Ai and xj = A0Aj of n-D simplex equals 60o, i.e.
∠xi,xj = 60o for any pair of the vectors xi and xj. Setting cos(∠xi,xj) = cos60o = ½, for the vectors spanning the simplex in n-D space and assuming xij ≥ 0, one can write:
x112 = 1
(M15)
x11⋅xi1 = 1/2 (for i = 1,…,n)
(M16)
x212
(M17)
+
x222
=1
x21⋅xi1 + x22⋅xi2 = 1/2
(M18)
x312
(M19)
+
x322
+
x332
=1
x31⋅xi1 + x32⋅xi2 + x33⋅xi3 = 1/2
(M20)
From eq. M15 we have x11 = 1. Putting it into (M16), we get xi1 = 0.5; among others, x21 = 1/2, and
then from the relation (M17) we get x22 =
From (M18) we get xi2 =
3 /2 ≅ 0.866.
3 /6 (= 3 /2/3 ≅ 0.289). Then we take the relations: (M19) and (M20) and
one obtains, by turns: x33 =
6 /3 ≅ 0.817, xi3 =
6 /12 (= 6 /3/4 ≅ 0.204). As we see (compare with
matrices (M1), (M2)), the co–ordinates of the points in the matrix (M5) are derivable with none relevance to geometrical properties of the related geometrical figures (i.e. equilateral triangle, tetrahedron). Such a circumstance can be exploited for calculation of further xij values using, by turns, the
relations:
x412 + x422 + x432 + x442 = 1
(M21)
x41xi1 + x42xi2 + x43xi3 + x44xi4 = 1/2
(M22)
and, generally,
∑k=1i xik2 = 1
(M23)
∑k=1m
(M24)
xikxjk = 1/2 where m = min(i,j)
One can introduce the notations: x11 = h1 = 1 = [½⋅(1+1)/1]1/2, x22 = h2 =
1/2
3 /2 = [½⋅(2+1)/2] , x33 = h3
= [½⋅(3+1)/3]1/2, ..., and xi1 = r1 = h1/(1+1), xi2 = r2 = h2/(2+1), xi3 = r3 = h3/(3+1), ... . Then the n-D simplex can be written in the form
5
X=
0 0

h1 0
 r1 h 2

 r1 r2
 ... ...

 r1 r2
0

0
0

0
... 

hn 
(M25)
(for 1 ≤ j ≤ n)
(M26)
0
...
0 ...
0 ...
h3 ...
... ...
r3 ...
where:
1/ 2
 j + 1

h j = 
 2⋅ j 
1/ 2
1  j +1


rj =
j + 1  2 ⋅ j 
1/ 2


1

= 
 2 j ( j + 1) 
(for j < i ≤ n)
(M27)
One can notice that hj (eq.M26) is the height of j-D symmetrical simplex and rj (eq.M27) is the radius of
j-D sphere (for j=2 – circle, for j=3 – ‘normal’ sphere) inscribed in j-D simplex. Moreover, we have the
relations:
h j = rj + R j
(M28)
R j = j ⋅ rj
(M29)
where Rj is the radius of j-D sphere circumscribed on j-D simplex.
R2
h2
r2
Fig.M5. Graphical presentation of r2 , h2 and R2 in 2-D space (equations M28 and M29)
1.3.2. Translation and reflection of the simplex
Simplex can translate and/or rotate in space. Particularly, the simplex defined by X (eq. M25), with the
centre at Ac = (r1,...,rn), can be parallely shifted to the centre Pc0 = (0,...,0) of the co–ordinate system.
This way, one obtains a new matrix Xtr
6
− r1

 R1
 0

 ...
 0

 0
Xtr =
− r2
− r3
... − rn−1
− r2
− r3
... − rn−1
R2
− r3
... − rn−1
...
...
...
...
0
0
0
0
...
R n −1
...
0
− rn 

− rn 
− rn 

... 
− rn 

R n 
(M30)
with Rj and rj (j = 1,...,n) as ones related in eq. M29. Reflection of particular vertices of the matrix Xtr in
the centre of co–ordinate system (0,...,0) gives the simplex described by the matrix
 r1

− R 1
 0
Xtr·(–I) = 
 ...
 0

 0
X* =
r2
r3
...
rn−1
r2
r3
...
rn−1
− R2
r3
...
rn−1
...
... ...
...
0
0
... − R n−1
0
0
...
rn 

rn 
rn 

... 
rn 

− R n 
0
(M31)
where I = [ δ ij ] – identity matrix.
The positions of the simplex presented in Fig. M6 (in 2-D space), obtained from (I) by translation (II)
and mirror reflection (III) in the origin of ordinate axis, are presented by matrices: X (eq. M31 for n = 2
and
Xtr =
-r1 -r2
R1 -r2
0 R2
=
-0.5 -0.289
0.5 -0.289
0
0.578
X* =
I
X2
=
0.5 0.289
-0.5 0.289
0 -0.578
II
A2
A0
r 1 r2
-R1 r2
0 -R2
X2
A1 X1
III
A2’
A0”
X2
A1”
X1
A0’
(M32)
A1’
X1
A2”
Fig. M6. The simplexes obtained after successive (II) translation and (III) reflection of the original simplex (I) in the center of co–ordinate axis in 2-D space; for details – see text.
2. Example of simplex planning
The principle of the simplex planning is exemplified by gravimetric determination of Cd, based on
the reaction of bromine with 8–hydroxyquinoline (oxine, HL).
7
Reagents and solutions
•
Solution of CdSO4⋅8/3H2O (m0 = 0.01272 g Cd/1g), standardised according to electrogravimetric and complexometric methods;
•
8–hydroxyquinoline HL (3.00%, m/v) in ethanol;
•
tartaric acid (0.220 g/ml);
•
NaOH (2.51 mol/l);
•
Doubly distilled water.
All reagents applied were of analytical purity grade.
Analytical procedure
The pipetted samples, ca. 5 ml of the standardised CdSO4 solution, weighed on analytical balance
(±0.1 mg), were taken for analysis. The solution was treated with tartaric acid solution and then with
NaOH solution. After dilution with water, the solution was treated with 8-hydroxyquinoline solution
added dropwise from burette, with defined rate of addition. The mixture was leaved for defined time on
the water-bath for re-crystallisation of the precipitate CdL2 at defined temperature, then filtered,
washed with water, dried to a constant mass at 105 oC and weighed.
Decision variables
Z1 – re–crystallisation time [min] of the precipitated CdL2;
Z2 – mass [g] of tartaric acid added in the solution;
Z3 – volume [ml] of NaOH added;
Z4 – volume [ml] of 8–hydroxyquinoline added;
Z5 – rate [ml/min] of 8–hydroxyquinoline addition;
Z6 – temperature [oC] of water bath where re–crystallisation of CdL2 occurs;
Z7 – volume [ml] of the solution before precipitation.
Matrix for initial simplex
The number of decision variables: n = 7 equals to the number of columns in the simplex matrix X.
Number of rows in the initial simplex: n+1 = 8.
 0

 1
0.5

0 .5
X = [xij] = 
0.5

0.5
0.5

0.5









0.289 0.204 0.158 0.129 0.764
0 

0.289 0.204 0.158 0.129 0.109 0.756 
0
0
0
0
0
0
0
0
0.866
0
0
0
0.289 0.817
0
0
0.289 0.204 0.791
0
0.289 0.204 0.158 0.775
0
0
0
0
0
0
0
0
0
0
0
0
A0
A1
A2
A3
A4
A5
A6
A7
(M33)
Criterion of optimisation
The criterion of optimisation suggested is the sum of squares
y(i) = 1/ Ni ⋅
N
∑ k =i 1 (mik − m0 )2 = (mi − m0 )2 + (1 − 1/ Ni ) ⋅ si2
(M34)
8
where the values:
N
mi = 1/ Ni ⋅ ∑ k =i 1 mik
(M35)
N
si2 = (Ni − 1) −1 ⋅ ∑ k =i 1 (mik − mi ) 2
(M36)
were calculated in the i–th simplex point Ai (eq. M33) on the basis of Ni results mik of measurements of
an intensive variable m (e.g. number of grams of an analyte contained in unit mass of solution); m0 is
the true value of this variable, obtained according to a reference method. The optimisation is terminated at the simplex point where
si
⋅ t(α, fi )
Ni
mi − m 0 <
where si =
(M37)
si 2 , t(α, fi ) is the critical t-value for Student t-test found for fi = Ni – 1 degrees of free-
dom on α = 0.95 probability level assumed. In our case we have
s
mi − m0 < i ⋅ 3.182 = 1.591⋅ si
4
(M38)
Values of natural variables Zj (j = 1,…,7) within the initial simplex
The value zij of j-th natural variable (Zj) at i-th point of the initial simplex (i = 0,…,n; j = 1,…,n) is defined by equation
zij = z0 j + x ij ⋅ ∆z j
(i = 0,1,...,7; j = 1,...,7)
(M39)
where i = 0,1,...,n enumerates successive experimental points related to the initial simplex X = [xij]
represented by the matrix M33. On the basis of pre-assumed values for z0j and ∆zj, one can calculate
the values zij of the natural variables Zj from Eq. M39. The value z0j is presented in row i=0 for variable
Zj in j-column in Table 1.
For example, the value z04 = 7.20 [ml] refers to variable Z4; the related value ∆z4 = 1.00 ml. From
M37 and M34 and Table 1, for i = 3 and j = 4 we have z34 = 8.00 + 0.817⋅1.00 ≅ 8.82 ml.
Table 1. The values for zij assumed in points of the initial (columns 1–7, rows 0–7) and evolving (columns 1–7, rows 8–10) simplexes. For all points, the values y(i) were obtained on the basis of Ni = 4
measurements.
j
i▼
0
1
2
3
4
5
6
7
8
9
10
∆zi ►
z0j
z1j
z2j
z3j
z4j
z5j
z6j
z7j
z8j
z9j
z10,j
1
2
3
4
5
6
7
60
60
120
90
90
90
90
90
90
90
90
90
1.00
3.00
3.00
3.87
3.29
3.29
3.29
3.29
3.29
2.55
2.91
2.80
1.00
8.00
8.00
8.00
8.82
8.20
8.20
8.20
8.20
8.47
8.34
7.59
1.00
7.20
7.20
7.20
7.20
7.99
7.36
7.36
7.36
7.56
7.46
7.69
1.00
1.00
1.00
1.00
1.00
1.00
1.78
1.13
1.13
1.30
1.21
1.40
10
60
60
60
60
60
60
67.5
61
62.5
62
63
50
100
100
100
100
100
100
100
138
111
65
93
y(i)⋅108
1.2
0.7
3.0
1.9
0.7
1.3
0.8
2.3
1.3
1.8
0.5
9
Experiments done within the initial simplex
The simplex optimisation procedure was realized in experiments performed in all points of the initial
simplex (rows 0,…,7). In each point Ai (i = 0,…,7), Ni = 4 measurements were done. The list of results
mik [g Cd/g] is presented in Table 2.
Table 2. M table with results of measurements done within the initial simplex. The results are recalculated on Cd contents [g] in 1.0000 g of the solution; j = 0,…,7; k = 1,…,4.
Ai
A0
A1
A2
A3
A4
A5
A8.
A7
mik
0.01260
0.01265
0.01256
0.01256
0.01265
0.01262
0.01262
0.01255
[g/g]
0.01263
0.01268
0.01252
0.01258
0.01267
0.01255
0.01268
0.01259
0.01257
0.01259
0.01249
0.01269
0.01260
0.01267
0.01261
0.01262
0.01268
0.01265
0.01267
0.01255
0.01264
0.01262
0.01264
0.01253
On the basis of the data in Table 2, the y(i) values were calculated (last column in Table 1). As results
from detailed calculations done on the basis of results in Table 2, the inequality M39 has not been
fulfilled within the initial simplex and then the evolution of the simplex was required.
Evolution of the simplex
At the first stage of the simplex evolution, a point Aw1 with the worst (w1) value, y(w1), for the object
function is indicated, w1∈ < 0, n > . The point Aw1 is replaced by a new point, An+1, obtained by reflection of Aw1 in the centre (Ac1) of the remaining points of the initial simplex, i.e. in n–1-D wall not containing the point Aw1. The point Aw1 is replaced by a new point, An+1, obtained by reflection of Aw1 in the
centre, Ac1(xc1), of the remaining n points of the initial simplex, i.e. in n–1-D wall not containing the
point Aw1. In other words, xc1 = ( x1 ,..., xn )c1 is the middle point between xw1 and xn+1, i.e.
T
(xw1 + xn+1)/2 = xc1 = 1/n ⋅ ∑i≠w1 xi
(M40)
After adding the term (1/n)⋅xw1 to both sides of (M40), we obtain the co–ordinates of the new point An+1
xn+1 = 2/n ⋅ ∑i=0n xi – (1 + 2/n) ⋅ xw1
(M41)
After measurement(s) done at the point An+1, the values y(i) are compared again within the new
(evolved) simplex (containing An+1 and not containing Aw1) and the point Aw2 with the worst value,
y(w2) is chosen. Then Aw2 is reflected in the centre of n–1-D wall of the new simplex (not containing
w2). This way, a new point An+2 is obtained and further operations are repeated again, in cyclic manner. At q–th reflection one can write
(xwq + xn+q)/2 = xcq = 1/n ⋅ ∑i≠wq(q) xi
(M42)
where , xcq = ( x1 ,..., xn )cq are co–ordinates of the centre of remaining (except the worst one) points of
T
the q–th evolving simplex.
Referring to the Table 1, the worst (w=w1) result within the initial simplex was obtained at the point
A2. During the simplex evolution, this point was reflected in the centre xc1 of the remaining points of
the starting simplex. Referring to Table 1, we have, for example,
x2c1 = (2⋅0 + 5⋅0.289)/7 = 0.2064
(0.866 + x82)/2 = 0.206 Æ x82 = – 0.454
z82 = 3.00 + (–0.454)⋅1 ≅ 2.55
10
x3c1 = (2⋅0 + 0.817 + 4⋅0.204)/7 = 0.233
(0 + x83)/2 = 0.233 Æ x83 = 0.466
z83 = 8.00 + 0.466⋅1 ≅ 8.47
At the point A8, the condition M38 has not been fulfilled as well, and further evolution of the simplex
was required. The worst y(w2) value within the new simplex was stated in the point A7. Further calculations referred to the new point A9 are exemplified below:
X2c2 = (2⋅0 + 4⋅0.289 + 1⋅(–0.454))/7 = 0.1004
(0.289 + x92)/2 = 0.1004 Æ x92 = –0.088,
z92 = 3.00 + (–0.088)⋅1 ≅ 2.91;
x3c2 = (2⋅0 + 0.817 + 3⋅0.204 + 0.467)/7 = 0.271
(0.204 + x93)/2 = 0.271 Æ x93 = 0.338
P93 = 8.00 + 0.338⋅1 ≅ 8.34
x7c2 = (6⋅0 + 0.216)/7 = 0.031
(0.756 + x97)/2 = 0.031 Æ x97 = –0.694,
z97 = 100 + (–0.694)⋅50 ≅ 65;
x2c3 = (2⋅0 + 3⋅0.289 + 1⋅(–0.454) + 1⋅(–0.088))/7 = 0.046
(0.289 + x10,2)/2 = 0.046 Æ x10,2 = –0.197
z10,2 = 3.00 + (–0.197)⋅1 ≅ 2.80
The inequality (M38) has not been fulfilled within the new simplex, as well. The worst result within the
new simplex has been stated at the point A3. Then we calculate, among others:
x3c3 = (2⋅0 + 3⋅0.204 + 0.467 + 0.338)/7 = 0.2024
(0.817 + x10,3)/2 = 0.2024 Æ x10,3 = –0.412,
z93 = 8.00 + (–0.412)⋅1 ≅ 7.59.
The steps ∆zj for the co–ordinates Zj (i.e. values for natural variables) at the points Ai (i=0,1,...) refereed to the initial (i=0,...,7) and evolving (i=8,9,10) simplex are specified in Table 1. Then the co–
ordinates of the new points: A8, A9 and A10 for evolving simplexes were calculated with use of the
coded variables xij, are specified in Table 3.
0.5 − 0.453 0.467 0.361 0.295 0.249 0.216
0.5 − 0.088 0.338 0.261 0.214 0.180 − 0.694
0.5 − 0.196 − 0.412 0.494 0.404 0.341 − 0.137
A8
A9
A 10
respectively.
The optimisation procedure has been terminated at the point A10, where mean Cd-contents mi =
0.01267 and variance si2 = 0.32⋅10-8 were found on the basis of 4 measurements and then the inequality (M38)
0.01267 – 0.01272 < 0.56⋅10–4⋅1.591 = 0.89⋅10–4
was fulfilled and
y(10) = (0.01267 – 0.01272)2 + (1 – 1/4)⋅0.32⋅10–8 = 0.49⋅10–8 ≅ 0.5⋅10–8
(see Table 1). It should be noticed that the y(i) (eq.M34) is not caused only by changes in Zi values; it
is partly ascribed to random errors.
11
Optimal prescription
On the basis of the optimization procedure, one can formulate the following prescription.
The pipetted samples, ca. 5 ml of the standardised CdSO4 solution, weighed on analytical balance
(±0.1 mg), were taken for analysis. The solution was treated with 2.80 g of tartaric acid and then with
7.9 ml of 2.5 mol/l NaOH solution. After dilution with water up to ca 90 ml, the solution was treated with
7.7 ml of 3% (m/v) 8-hydroxyquinoline solution (added dropwise from burette, with the rate 1.4 ml/min.
The mixture was leaved for 90 min. on the water-bath for re-crystallisation of the precipitate CdL2 at
temperature 63 oC, then filtered, washed with water, dried to a constant mass at 105 oC and weighed.
2. Description of computer program – an example
All the calculations can be performed according to a very simple computer program written in MATLAB; such an option facilitates the optimisation procedure. The program is based on transformations
done on the related tables (matrices). The results obtained according to the computer program can be
compared with ones presented in the paper.
The principle of the program is not confined, however, to the example considered in this paper. After
minute changes, the program can be adopted to another optimisation purposes, made according to
simplex method, for any number (n) of factors considered in the optimisation. If a user aims to resolve
its own optimisation problem, he should follow the procedure analogous to one specified below.
After taking the file simplex.m, it is inserted in the corresponding MATLAB catalogue. Double clicking
opens the file in the text editor. Then the „Debug” and „Run” commands are chosen from Menu
placed at the top of the screen.
In order to follow the calculations, the experimental data, arranged conveniently for user’s needs, can
be inserted in the commands line.
Starting values, Zi0, for the variables (factors) Zi and their steps (∆Zi) are chosen arbitrarily. They are
taken, at least partially, from analytical prescription (if attainable). Generally, some limitations put on
temperature (possibility of phase change) and concentrations (limited solubility) should also be taken
into account - both within the initial and evolving simplexes.
At the start for optimisation, the number (n) of the optimising factors should be declared. On this basis,
the (n+1) × n table, i.e. matrix X(n+1,n) with n columns and n+1 rows, with coded variables (xij) is
formulated. For example, putting n = 7, one gets the matrix X(7,8), with 7 columns and 7+1=8 rows.
The rows, numbered 1,...,8, define particular vertices (A0, A1,...,A7) of the starting simplex. The values
in columns of the matrix X = [xij] with coded variables xij refer to j-th factor, assumed within the starting
simplex.
Then the starting values Zi0 for real variables Zi are inserted. In our case (see Table 1), in the line of
commands we write, successively, the values: 60.0, 3.0, 8.0, 7.20, 1.0, 60.0, 100.0; the values introduced are separated by comma or spaced. The values for steps (∆Zi) for the corresponding variables
Zi are introduced in the same sequence as the Zi0–values assumed in the preceding line.
Referring to our case, in the line of commands we write: 60.0, 1.0, 1.0, 1.0, 1.0, 10.0, 50.0. Then, on
the basis of eq. (24), the program calculates, in natural units, the co–ordinates for the corresponding
12
simplex points (vertices). This way, the matrix Z=[Zij] equidimenssional with the matrix X = [xij] of
coded variables, is formulated.
The true contents (m0, g/g) of Cd in the solution tested is inserted; we write the number 0.01272 in the
command line. Next, the value for confidence level (α, %) due to Student’s t–test is assumed; we write
the number 95 in the line of commands. In the program, equal number of measurements (Ni = 4) was
assumed at each (i-th) simplex point; we write the number 4 in the line of commands. On the basis of
the α and Ni values, the program calculates critical values t(α,f) for the Student’s t–test, t(α,f) = t(0.95,
3) = 3.182 (f = 4 – 1 = 3).
In experimental part of the optimisation procedure, the measurements were performed under the conditions provided by the Table 1. In our case, 8 x 4 = 32 individual measurements of Cd–contents (mik,
g/g) were done.
The results of measurements, specified in Table 2 (table M), obtained in particular points of the initial
simplex, are written successively and spaced, in the line of commands :
0.01260
0.01265
0.01256
0.01256
0.01265
0.01262
0.01262
0.01255
0.01263
0.01268
0.01252
0.01258
0.01267
0.01255
0.01268
0.01259
0.01257
0.01259
0.01249
0.01269
0.01260
0.01267
0.01261
0.01262
0.01268
0.01265
0.01267
0.01255
0.01264
0.01262
0.01264
0.01253
After inserting the results related to i–th point of the simplex, press ENTER and then the next line is
written.
Summarising, the table M (4×8) of experimental results is thus obtained. The results in each column
are referred to particular measurements done at all eight points of the starting simplex. Then mean
values, mmean(i), for results (m) found in particular simplex points and standard deviations ( s(i) ) are
calculated and collected in the corresponding tables, denoted as mmean(1x8) and s(1x8). It enables
the y(i) values ( table y(1x8) ) to be calculated in particular points of the initial simplex, compare with
Table 1 (last column, rows 0 ÷ 7). The y(i) value is the criterion of quality of results obtained in i–th
point of the simplex. Low y(i) values are obtained for results characterised by relatively good accuracy
and good precision.
In order to choose whether the condition given by inequality (M37) is fulfilled, two next tables, named
as:
absbias =  mi – m0
ref = ( s i / N i )⋅t(α, fi)
were prepared (absbias – abbr. for absolute bias, ref – abbr. for reference value).
Then the corresponding elements of the tables (absbias and ref) are compared. If the element of the
table absbias is lower than the corresponding element of table ref, then optimal conditions are found at
this point.
Referring to the optimisation procedure illustrated in Table 1, one can state that the inequality (M37)
was not fulfilled at any point (i = 0,1,...,7) of the initial simplex. Then we were forced to indicate the
13
point (Aw1) characterised by greatest y(w1) (i.e. the worst) value; w1 ∈ < 0, 7 >. From detailed inspection (Table 1), we have stated that the worst result, y(3) = y(w1) = 3.0⋅10-8, occurs at i = 2, i.e. at the
third simplex point (A2).
In further part of the optimisation procedure, the point Aw1, referred to the vector xw1, is reflected in the
centre (Ac1) of the remaining points of the starting simplex. This operation is done according to eq. 20.
This way, the new point, denoted in Table 1 as A8, is obtained and the starting (‘old’) simplex is replaced by the new (evolved) simplex containing A8 and not containing A2 = Aw1. In the new (updated)
matrix X, referred to the evolved simplex, the row referred to the worst result is replaced by the row
referred to the point A8 and the (updated) matrix Z of natural variables is determined. The updated
matrix involves the row with (coded or natural) co-ordinates of the point A8 and the program displays
the information:
“Optimal values for factors were not found within the scope of the initial simplex. The worst result was
found in the point no. 3. Do further measurements at the following values for factors assumed at the
point A8:
90.0000
2.5464
8.4666
7.5614
1.2951 62.4939 110.7990 ”
where the measurements were done. The values for the parameters assumed in measurements were
rounded (reasonably) to three significant numbers, i.e.
90.0
2.55
8.47
7.56
1.30 62.5 111
After 4 measurements done at the point A8, the following results were obtained:
A8
m8k
0.01260
0.01255
0.01263
0.01268
This column of numbers is inserted into the table M as the column inscribed to A2 and mk2. This operation facilitates the calculation procedure and any additional activities involved with assigning the numbers designed for calculation are not needed.
These values were inserted in the command line as:
0.01260 0.01255 0.01263 0.01268
On this basis, the tables: M, mmean, s, y, absbias and ref are updated; e.g. the value for y(3) ≈
3.0⋅10-8 is updated by the new value y(8) ≈ 1.3⋅10-8.
Then the absbias and ref values referred to the point A8 are compared, as previously. It was stated
that the inequality (M38) was not fulfilled. This circumstance required further evolution of the simplex.
For this purpose, the point A7 with the worst value y(8) ≈ 2.3⋅10-8 has been reflected in the centre xc2 of
the remaining points and coordinates of the new point, A9, are found. Further activities were the same
as above; they can be formulated as a loop of experimental and computational activities, considered
repeatedly and alternately. After each successive step, where optimal values required from the viewpoint of relation (M38) were not obtained, the program displays the information similar to one specified
above.
14
On the basis of measurements done on particular steps, the new numerical data evidenced every time
in the corresponding tables (i.e. M, mmean, s, y, absbias and ref) are displayed. All the procedure is
terminated at the point where the criterion of optimisation (relation M38) is reached. In our case, the
inequality (M38) was attained at the point A10.
As we see, the measurements at the points A9 and A10 were also required. At the point A9, the program displays the following information
“Optimal values for factors were not found within the scope of the evolved simplex. The worst result
was found in the point no.8. Do further measurements at the following values for factors assumed at
the point A9:
90.0000
2.9116
8.3374
7.4614
1.2134 61.8036 65.2890”
(see table 1, row no.9).
After 4 measurements done at the point A9, the following results were obtained:
A9
m9k
0.01257
0.01264
0.01260
0.01255
and written in the line of commands as :
0.01257 0.01264 0.01260 0.01255
At the point A9, the program displays the following information
“Optimal values for factors were not found within the scope of the evolved simplex.
The worst result was found in the point no.4. Do further measurements at the following values for factors assumed at the point A10:
90.0000
2.8039
7.5882
7.6942
1.4035 63.4101 93.1680
After 4 measurements done at the point A10, the following results were obtained:
A10
m10,k
0.01261
0.01274
0.01268
0.01264
and written in the line of commands as :
0.01261 0.01274 0.01268 0.01264
At the point A10, the optimal values were reached and the program displayed the following information:
” The following optimal values were searched:
Zopt =
90.0000
2.8039
7.5882
7.6942
1.4035 63.4101 93.1680”
(see Table 1, row no. 10).
Ultimately, the optimal conditions were found after three evolving procedures realised on the initial
simplex.
15
Remarks
The optimisation procedure presented above requires some comments. It may happen that the condition (M38) is fulfilled in i-th point of the starting simplex, at i < n. In this case, the optimisation procedure can be terminated at this point, without the necessity to do further measurements in the remaining points of the simplex. So the measurements done factually in the remaining points of the starting
simplex provide a kind of redundant information. However, the measurements done in the remaining
point tended to check out whether better results for y(i) are obtainable.
3. Printout of the computer program
clear;
clc;
fnd=0;
n=input('Give the number of factors(n): ')
X=zeros(n+1,n);
Z0=zeros(n+1,n);
DZ=zeros(n+1,n);
% Applying the formulae for hj (eq. 10) and rj (eq. 11), formulate the matrix X (eq. 8) for the n-value
assumed
for j=1:1:n
X(j+1,j)=sqrt((j+1)/(2*j));
end
for u=1:1:(n-1)
for j=1:1:(n-u)
X(j+u+1,j)=sqrt(1/(2*j*(j+1)));
end
end
display(strvcat('Input the number (N) of experimental points at each (i-th) point of the simplex',...
'Equal number (Ni = N) of measurements at each (i-th) point is assumed'))
Ni=input('Input the number of measurements (N) at each point of the simplex: ')
display('Write: 0.01272')
m0=input('Give true value: ')
display('Write 95')
doft=input('Give confidence level in [%]: ')
doft=doft/100;
alpha=1.0-doft;
doft=(1-alpha/2);
X;
% Formulation of the matrix with starting values (Z0j, j = 1,...,n) of natural variables
display('Write: 60.0, 3.0, 8.0, 7.20, 1.0, 60.0,100.0')
Z0j=input('Give starting value for natural variables: ','s')
Z0j=eval(strcat('[',Z0j,']'))
% Formulation of the matrix with the steps DZj (j = 1,...,n) for the starting values Z0j
display('Write: 60.0, 1.0, 1.0, 1.0, 1.0, 10.0, 50.0')
DZj=input('Give step value for natural variables: ','s')
dZj=eval(strcat('[',DZj,']'))
for a=1:1:(n+1)
Z0(a,:)=Z0j;
DZ(a,:)=dZj;
end
Z0;
DZ;
% From matrices: X, Z0 and DZ, the components of the matrix Z (eq.18) are
% calculated
16
Z=Z0 +X.*DZ
for w=1:1:(n+1)
display(strvcat(' Write in A0 : 0.01260 0.01263 0.01257 0.01268',...
' Write in A1 : 0.01265 0.01268 0.01259 0.01265',...
' Write in A2 : 0.01256 0.01252 0.01249 0.01267',...
' Write in A3 : 0.01256 0.01258 0.01269 0.01255',...
' Write in A4 : 0.01265 0.01267 0.01260 0.01264',...
' Write in A5 : 0.01262 0.01255 0.01267 0.01262',...
' Write in A6 : 0.01262 0.01268 0.01261 0.01264',...
' Write in A7 : 0.01255 0.01259 0.01262 0.01253'))
display(strcat('Input the results of measurements at the point A',num2str(w-1),': '))
data=input('Input the results of measurements:(in line, separated by comma or spaced): ','s')
data=eval(strcat('[',data,']'))
data=data'
M(:,w)=data
end
% Formulation of the matrix mmean with mean values referred to results
% of measurements done in the simplex points
mmean=sum(M,1)./Ni
% Formulation of the matrix with standard deviations referred to results
% of measurements done in the simplex points
s=std(M,0,1)
% Formulation of the matrix for object function (y, eq. 15) values
y=(mmean-m0).^2 + (1-1/Ni).*(s.^2)
% Formulation of matrices: absbias and ref
ref=abs(mmean-m0)
absbias=s./sqrt(Ni).*tinv(doft,Ni-1)
% Check out whether the inequality (17) is fulfilled within the initial simplex
for i=1:1:n+1
if ref(1,i) < absbias(1,i)
display('The following optimal values were searched:')
Zopt=Z(i,:);
return
end
end
% Searching the worst point (w) in the matrix y
[C,I] = max(y,[],2)
nbpn=num2str(I);
% Mirror reflection of the worst point (w) in the centre of remaining points of current simplex
% Calculation of co-ordinates of the new simplex point (eq. 20) and inserting it in the 'old' matrix X
X(I,:)=(2/n).*(sum(X,1)) - X(I,:).*(1+2/n)
% Calculation of co-ordinates of the new matrix Z with natural variables (eq. 18),
% involving the new simplex point, put for the worst point
Z(I,:)=Z0(I,:)+ X(I,:).*DZ(I,:)
% Check out whether the inequality (17) is fulfilled within the new simplex
for i=1:1:n+1
if ref(1,i) < absbias(1,i)
display('The following optimal values were searched:')
Zopt=Z(i,:);
return
end
end
% Replacing, in matrix M, the results in the worst point (w) by results of measurements obtained in
the new point
if fnd==0
17
display(strvcat('Optimal values for factors were not found within the scope of the initial simplex. '...
,strcat('The worst result was found in the point no.',nbpn,'.')...
,strcat('Do further measurements at the following values for factors assumed at the point
A',num2str(n+1))) );
Z(I,:)
end
nbpn0=n+2;
while fnd==0
Z=Z0 +X.*DZ;
display(strcat('Input the results of measurements at the point A',num2str(nbpn0-1),': '))
display(strvcat(' Write in A8 : 0.01260 0.01255 0.01263 0.01268',...
' Write in A9 : 0.01257 0.01264 0.01260 0.01255',...
' Write in A10 : 0.01261 0.01274 0.01268 0.01264'))
newdata=input('Input the results of measurements:(in line, separated by comma or spaced):','s')
newdata=eval(strcat('[',newdata,']'))
newdata=newdata'
M(:,I)=newdata
% Updating the matrix mmean with mean values...
mmean=sum(M,1)./Ni
% ... and the matrix s of standard deviations
s=std(M,0,1)
% Calculation of the matrix y for n+1 points of the new (evolved) simplex
y=(mmean-m0).^2 + (1-1/Ni).*(s.^2)
% Updating the matrices: ref and absbias (compare with relation 17)
ref=abs(mmean-m0)
absbias=(s./sqrt(Ni)).*tinv(doft,Ni-1)
% Check out whether a simplex point exists where the inequality (17) is fulfilled
for i=1:1:n+1
if ref(1,i) < absbias(1,i)
Zopt=Z(i,:);
fnd=1;
end
end
% Searching the worst point (with maximal y-value) in the current simplex
[C,I] = max(y,[],2)
nbpn=num2str(I);
nbpn0s=num2str(nbpn0);
% The mirror reflection of the worst point in the current simplex
% Calculation of co-ordinates of the new simplex point and putting them
% for co-ordinates of the worst point (formation of the updated matrix X)
X(I,:)=(2/n).*(sum(X,1)) - X(I,:).*(1+2/n)
% Calculation of updated matrix Z (see eq. 18), with the new point put for the worst point
Z(I,:)=Z0(I,:)+ X(I,:).*DZ(I,:)
if fnd==0
display(strvcat('Optimal values for factors were not found within the scope of the evolved simplex. '...
,strcat('The worst result was found in the point no.',nbpn,'.')...
,strcat('Do further measurements at the following values for factors assumed at the point
A',nbpn0s)) );
Z(I,:)
end
nbpn0=nbpn0+1;
end
display('The following optimal values were searched :')
Zopt
18