Optimization techniques

1. optimization techniques
pradeep kumar c
roll no: 09
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2. optimization
“An art, process, or methodology of making
something (a design, system, or decision) as
perfect, as functional, as effective as possible.”
2

3. seminar outline:
Introduction
Key term used in in optimization
Optimization Parameters
Types of experimental designs
Applied Optimization Methods
References
3

4. objective of
pharmaceutical
optimization
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5. ADVANTAGES
•Yield the “best solution” within the domain
of study.
•Require fewer experiments to achieve an
optimum formulation.
• Optimization makes the perfect formulation
& reduce the cost
•Can trace and rectify “problem” in
a remarkably easier manner
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6. Key term used in optimization process
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7. Optimization Parameters
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1.Problem types:
Constraints
Example-Making hardest tablet but should disintegrate within 15 mins
( Constraint)
Unconstraint
Example: Making hardest tablet ( Unconstraint)
2. Variables:
Independent variable- E.g. - mixing time for a given process step.
granulating time.

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Dependent variables, which are the responses or the characteristics of
the in process material Eg. Hardness of the tablet.
Higher the number of variables, more complicated will be the
optimization process.
There should be a relationship between the given response and the
independent variable, and once this relationship is established , a
response surface is generated.
From response surface only, we find the points which will give
desirable value of the response.

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Response surface representing the relationship between
the independent variables X
1
and X
2
and the dependent
variable Y.

10. 10
Classic Optimization
•It involves application of calculus to basic problem for
maximum/minimum of a function.
•Limited applications
Applicable for problems that are not too complex and
that do not involve more than two variables
•For more than two variables graphical representation is impossible
•It is possible mathematically , but very involved ,making use of partial
derivatives , matrics ,determinants & so on.

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Contour Plot
GRAPH REPRESENTING THE
RELATION BETWEEN
THE RESPONSE VARIABLE AND
INDEPENDENT VARIABLE

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Using calculus the graph obtained can be
Y = f (x)
We can take derivative ,set it equal to zero & solve for x to obtain the maximum or
minimum .
When the relation for the response y is given as the function of two independent
variables,x
1
&X
2
Y = f(X
1
, X
2
)
The above function is represented by contour plots on which the axes represents the
independent variables x
1
& x
2

13. Flow Chat Of Optimization
13

14. •The classic calculus methods apply basically to unconstrained
problems . but in pharmacy all problem are constrained
•Deming and king presented a general flowchart.
•Involve the effect on a real system of changing some input
(some factor or variable) is observed directly at the output
(one measures some property), and that set of real data is used
to develop mathematical models.
•The responses from the predictive models are then used for
optimization
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15. TYPES OF EXPERIMENTAL
DESIGN
Completely randomized designs
Randomized block designs
Factorial designs
Full
Fractional
Response surface designs
Central composite designs
Box-Behnken designs
Three level full factorial designs 15

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Completely randomized Designs
 These designs compares the values of a response variable
based on different levels of that primary factor.
 For example ,if there are 3 levels of the primary factor
with each level to be run 2 times then there are 6 factorial
possible run sequences.
Randomized block designs
For this there is one factor or variable that is of primary
interest.
To control non-significant factors, an important technique
called blocking can be used to reduce or eliminate the
contribution of these factors to experimental error.

17. 17
Factorial Design
These are the designs of choice for simultaneous determination
of the effects of several factors & their interactions.
Symbols to denote levels are:
(1)- when both the variables are in low concentration.
a- one low variable and second high variable.
b- one high variable and second low variable
ab- both variables are high.
•Factorial designs are optimal to determined the effect of pressure
& lubricant on the hardness of a tablet
•Effect of disintegrant & lubricant conc . on tablet dissolution .
•It is based on theory of probability and test of significance.

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It identifies the chance variation ( present in the process due to accident) and
the assignable variations ( which are due to specific cause.)
Factorial design are helpful to deduce IVIVC.
IVIVC are helpful to serve a surrogate measure of rate and extent of oral
absorption.
BCS classification is based on solubility and permeability issue of drugs,
which are predictive of IVIVC.
BCS Class Solubility Permeability IVIVC
I High High Correlation( if
dissolution is rate
limiting)
II Low High IVIVC is expected
III High Low Little or no IVIVC
IV low Low Little or no IVIVC
BCS classification and its expected outcome on IVIVC for Immediate release formulation

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Factorial design
Full
• Used for small set of factors
Fractional
• It is used to examine multiple factors efficiently with fewer runs than
corresponding full factorial design
Types of fractional factorial designs
Homogenous fractional
Mixed level fractional
Box-Hunter
Plackett - Burman
Taguchi
Latin square

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Homogenous fractional
Useful when large number of factors must be screened
Mixed level fractional
Useful when variety of factors needed to be evaluated for
main effects and higher level interactions can be assumed to be
negligible.
Ex-objective is to generate a design for one variable, A, at 2
levels and another, X, at three levels , mixed &evaluated.
Box-hunter
Fractional designs with factors of more than two levels can be
specified as homogenous fractional or mixed level fractional

21. Plackett-Burman
It is a popular class of screening design.
These designs are very efficient screening designs
when only the main effects are of interest.
These are useful for detecting large main effects
economically ,assuming all interactions are negligible
when compared with important main effects
Used to investigate n-1 variables in n experiments
proposing experimental designs for more than seven
factors.
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Taguchi
 It is similar to PBDs.
 It allows estimation of main effects while minimizing variance.
Taguchi Method treats optimization problems in two categories,

[A] STATIC PROBLEMS :Generally, a process to be optimized has several
control factors which directly decide the target or desired value of the output.
[B] DYNAMIC PROBLEMS : If the product to be optimized has a signal
input that directly decides the output,
Latin square
They are special case of fractional factorial design where there is one
treatment factor of interest and two or more blocking factors

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Signal-to-Noise ratios (S/N), which are log functions of desired output

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•We can use the Latin square to allocate treatments. If the rows of the
square
represent patients and the columns are weeks, then for example the second
patient,in the week of the trial, will be given drug D. Now each patient
receives all five drugs, and in each week all five drugs are tested.

A B C D E
B A D E C
C E A B D
D C E A B
E D B C A

25. Response surface designs
•This model has quadratic form
 γ =β
0
+ β
1
X
1
+ β
2
X
2
+….β
11
X
1
2
+β
22
X
2
2
•Designs for fitting these types of models are
known as response surface designs.
•If defects and yield are the outputs and the
goal is to minimize defects and maximize
yield
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Two most common designs generally used in this response
surface modeling are
Central composite designs
Box-Behnken designs
Box-Wilson Central Composite Design
This type contains an embedded factorial or fractional factorial
design with centre points that is augmented with the group of ‘star
points’.
These always contains twice as many star points as there are
factors in the design

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The star points represent new extreme value (low & high) for
each factor in the design
To picture central composite design, it must imagined that
there are several factors that can vary between low and high
values.
Central composite designs are of three types
Circumscribed(CCC) designs-Cube points at the corners of
the unit cube ,star points along the axes at or outside the cube
and centre point at origin
Inscribed (CCI) designs-Star points take the value of +1 & -1
and cube points lie in the interior of the cube
Faced(CCI) –star points on the faces of the cube.

28. 28Generation of a Central Composite Design for Factors

29. Box-Behnken design
•Box-Behnken designs use just three levels of each factor.
• In this design the treatment combinations are at the midpoints
of edges of the process space and at the center. These designs
are rotatable (or near rotatable) and require 3 levels of each
factor
These designs for three factors with circled point appearing at
the origin and possibly repeated for several runs.
It’s alternative to CCD.
The design should be sufficient to fit a quadratic model , that
justify equations based on square term & products of factors.
Y=b
0
+b
1
x
1
+b
2
x
2
+b
3
x
3
+b
4
x
1
x
2
+b
5
x
1
x
3
+b
6
X
2
X
3
+b
7
X
1
2 +
b
8
X2
2+
b
9
X
3
2
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30. 30A Box-Behnken Design

31. Three-level full factorial designs
It is written as 3
k
factorial design.
It means that k factors are considered each at 3 levels.
These are usually referred to as low, intermediate & high
values.
These values are usually expressed as 0, 1 & 2
The three level for a continuous factor facilitates investigation
of a quadratic relationship between the response and each of
the factors
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32. Applied optimizAtion
32

33. 1. EVOP METHOD
 Make very small changes in formulation repeatedly.
The result of changes are statistically analyzed.
 If there is improvement, the same step is repeated until
further change doesn’t improve the product.
Where we have to select this technique?
This technique is especially well suited to a production situation.
The process is run in a way that is both produce a product that
meets all specifications and (at the same time) generates
information on product improvement.
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34. Advantages:
Generates information on product development.
Predict the direction of improvement.
Help formulator to decide optimum conditions for the
formulation and process.
Limitation
More repetition is required
Time consuming
 Not efficient to finding true optimum
Expensive to use.
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35. Example: In this example, A formulator can changes the
concentration of binder (no of experiment is done) and get the
desired hardness.
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36. 2.SIMPLEX METHOD
It was introduced by Spendley et.al.
A simplex is a geometric figure, defined by no. of points or
vertices equal to one more than no. of factors examined.
Once the shape of a simplex has been determined, the
method can employ a simplex of fixed size or of variable
sizes that are determined by comparing the magnitudes of
the responses after each successive calculation
It is of two types:
A. Basic Simplex Method
B. Modified Simplex Method.
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The simplex method is especially appropriate when:
Process performance is changing over time.
More than three control variables are to be changed.
The process requires a fresh optimization with each new lot of
material.
The simplex method is based on an initial design of k+1,
where k is the number of variables. A k+1 geometric figure in
a k-dimensional space is called a simplex. The corners of this
figure are called vertices.

38. Basic Simplex Method
It is easy to understand and apply. Optimization begins with
the initial trials.
 Number of initial trials is equal to the number of control
variables plus one.
These initial trials form the first simplex.
The shapes of the simplex in a one, a two and a three variable
search space, are a line, a triangle or a tetrahedron
respectively.
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39. Rules for basic simplex:
The first rule is to reject the trial with the least favorable
value in the current simplex
The second rule is never to return to control variable
levels that have just been rejected.
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40. Modified simplex method
It was introduced by Nelder-Mead in 1965.
It can adjust its shape and size depending on the response in
each step. This method is also called the variable-size
simplex method.
Rules:
1.Contract if a move was taken in a direction of less favorable
conditions
2.Expand in a direction of more favorable conditions.
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41. Example
•Special cubic simplex design for a three component mixture.
Each point represent a different formulation SA = stearic acid;
DCP= dicalcium phosphate, ST= starch
•Constraint : with the restriction that the sum of their total
weight must equal to 350 mg, 50 mg = active ingredient
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42. Example
•Development of an analytical method (a continuous flow
analyzer) by Deming and king.
•The two independent variable show the pump speeds for the
two reagents required in the analysis reaction.
•The initial simplex is represented by the lowest triangle; the
vertices represent the Spectrophotometric response.
•The strategy is to move toward a better response by moving
away from the worst response 0.25, conditions are selected at
the vortex 0.6 and indeed, improvement is obtained.
•One can follow the experimental path to the optimum 0.721.
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43. Spectrophotometric response at given wavelength
43

44. 3.LAGRANGIAN METHOD
It represents mathematical techniques.
It is an extension of classic method.
Applied to a pharmaceutical formulation and processing.
This technique require that the experimentation be completed
before optimization so that the mathematical models can be
generates.
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45. •Steps involved:
1. Determine the objective function.
2. Determine the constraints.
3. Change inequality constraints to equality constraints.
4. Form the Lagrange function F:
a. one Lagrange multiplier λ for each constraint
b. one slack variable q for each inequality constraint.
5. Partially differentiate the Lagrange function for each
variable and set derivatives equal to zero
6. Solve the set of simultaneous equations.
7. Substitute the resulting values into objective function45

46. •Where we have to select this technique?
This technique can applied to a pharmaceutical formulation
and processing.
•Advantages :
lagrangian method was able to handle several responses or
dependent variables
•Disadvantages:
Although the lagrangian method was able to handle several
responses or dependent variables, it was generally limited to
two independent variables
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47. Example
Optimization of a tablet.
 Phenyl propranolol(active ingredient)-kept constant
X1 – disintegrate (corn starch)
X2 – lubricant (stearic acid)
X1 & X2 are independent variables.
Dependent variables include tablet hardness, friability
,volume, invitro release rate e.t.c..,
It is full 3
2
factorial experimental design.
Nine formulation were prepared
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48. PPolynomial models relating the response variables to
independents were generated by a backward stepwise
regression analysis program.
Y= B
0
+B
1
X
1
+B
2
X
2
+B
3
X
1
2
+B
4
X
2
2
+B
5
X
1
X
2
+B
6
X
1
2
X
2

+ B
7
X1

X
2
2
+B
8
X
1
2
X
2
2....................(1)

Y – Response

B
i
– Regression coefficient for various terms containing
the levels of the independent variables.
X – Independent variables.
one equation is generated for each response or dependent
variable.
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49. Formulation
no,.
Drug Dicalcium
phosphate
Starch Stearic acid
1 50 326 4(1%) 20(5%)
2 50 246 84(21%) 20
3 50 166 164(41%) 20
4 50 246 4 100(25%)
5 50 166 84 100
6 50 86 164 100
7
8
9
50
50
50
166
86
246
4
84
164
180(45%)
180
180
Tablet formulations
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50. Tablet formulations
Constrained optimization problem is to locate the levels of stearic acid(x
1
)
and starch(x
2
).
This minimize the time of invitro release(y
2
),average tablet volume(y
4
),
average friability(y
3
)
To apply the lagrangian method, problem must be expressed
mathematically as follows
Y
2
= f
2
(X
1
,X
2
)-invitro release…………(2)
Y
3
= f
3
(X
1
,X
2
)<2.72 %-Friability………..(3)
Y
4
= f
4
(x
1
,x
2
) <0.9422 cm
3
…………..(4)
 5≤X1 ≤45………(5)
1≤X2 ≤41……….(6)
 Equation (5) and (6) serve to keep the solution within the experimental
range.
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51. Contour plot
51
Hardness of tablet Dissolution time (t
50
%)

52. Contour plot
•C) Feasible solution space indicate by crosshatched area
52

53. Unlike the Lagrangian method, do not require differentiability
of the objective function.
The response surface is searched by various methods to find
the combination of independent variables yielding an
optimum.
 Used for more than two independent variables.
It take five independent variables into account and is computer
assisted.
4.SEARCH METHOD
53

54. Example: Optimization of tablet formulation
The experimental design used was a modified factorial
Independent Variables Dependent Variables
X1 = Diluents ratio Y1 = Disintegration time
X2= Compression force Y2= Hardness
X3= Disintegrant levels Y3 = Dissolution
X4= Binder levels Y4 = Friability
X5 = Lubricant levels Y5 = Porosity
54

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56. •The first 16 trials are
represented by +1 and -1.
•The remaining trials are
represented by a -1.547,
zero or 1.547
•The data were subjected
statistical analysis, followed
by multiple regression
analysis
•The type of predictor
equation used in this design
is a second-order
polynomial:
56

57. For optimization itself , two major steps were used:
Feasibility search
For e.g the constraints in table were fed into the computer
and were relaxed ones at a time until a solution was found.
Grid search
Experimental range is divided into a grid of specific size
and methodically searched.
From an input of the desired criteria, the program prints out
all points (formulation) that satisfy the constraints.
57

58. 1. Select a system
2. Select variables:
a. Independent
b. Dependent
3. Perform experimens and test product.
4. Submit data for statistical and regression analysis
5. Set specifications for feasibility program
6. Select constraints for grid search
7. Evaluate grid search printout
8. Request and evaluate:.
a.“Partial derivative” plots, single or composite
b. Contour plots
58

59. 5.Canonical analysis
•It is a technique used to reduce a second order regression
equation.
•This allows immediate interpretation of the regression
equation by including the linear and interaction terms in
constant term.
•It is used to reduce second order regression equation to an
equation consisting of a constant and squared terms as
follows-
Y = Y
0
+λ
1
W
1
2
+ λ
2
W
2
2
+..
2variables=first order regression equation.
3variables/3level design=second order regression equation.
59

60. 60
In canonical analysis or canonical
reduction, second-order regression
equations are reduced to a simpler
form by a rigid rotation and translation
of the response surface axes in
multidimensional space, as
for a two dimension system.

61. References:
•Schwarts J. B., et al., Optimization techniques in
pharmaceutical formulation and processing, in: Banker G. S.,
et al. (eds), Modern pharmaceutics, Marcel Dekker Inc., 4th
edition (revised and expanded), vol- 121, 607-620, 2005.
•Jain N. K., Pharmaceutical product development, CBS
publishers and distributors, 1st edition,297-302, 2006.
•Cooper L. and Steinberg D., Introduction to methods of
optimization, W.B.Saunders, Philadelphia, 1970.
•Bolton. S., Stastical applications in the pharmaceutical
science, Varghese publishing house,3rd edition, 223.
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62. 62
•Optimization techniques in pharmaceutical industry: A
Review , Journal of Current Pharmaceutical Research 2011; 7
(1): 21-28
•Fundamentals of Biostatistics. By KHAN and KHANUM.
•Google search engine, www.Google.co.in

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