General Introduction to Design of Experiments (DOE)

Experimental domain: the experimental ‘area’ that is investigated (defined by the variation of the experimental variables). Factors: experimental variables that can be changed independently of each other Independent Variables: same as factors Continuous Variables: independent variables that can be changed continuously Discrete Variables: independent variables that are changed step-wise, e.g., type of solvent. Responses: the measured value of the result(s). from experiments Residual: the difference between the calculated and the experimental result


Introduction
Experimental design and optimization are tools that are used to systematically examine different types of problems that arise within, e.g., research, development and production.It is obvious that if experiments are performed randomly the result obtained will also be random.Therefore, it is a necessity to plan the experiments in such a way that the interesting information will be obtained.

Terminology
Experimental domain: the experimental 'area' that is investigated (defined by the variation of the experimental variables).Factors: experimental variables that can be changed independently of each other Independent Variables: same as factors Continuous Variables: independent variables that can be changed continuously Discrete Variables: independent variables that are changed step-wise, e.g., type of solvent.Responses: the measured value of the result(s).from experiments Residual: the difference between the calculated and the experimental result

Empirical models
It is reasonable to assume that the outcome of an experiment is dependent on the experimental conditions.This means that the result can be described as a function based on the experimental variables [2] , Y= (f) x.The function (f) x. is approximated by a polynomial function and represents a good description of the relationship between the experimental variables and the responses within a limited experimental domain.Three types of polynomial models will be discussed and exemplified with two variables, x1 and x2.

Screening experiments
In any experimental procedure, several experimental variables or factors may influence the result.A screening experiment is performed in order to determine the experimental variables and interactions that have significant influence on the result, measured in one or several responses. [3] Factorial design [4] In a factorial design the influences of all experimental variables, factors, and interaction effects on the response or responses are investigated.If the combinations of k factors are investigated at two levels, a factorial design will consist of 2k experiments. In le 1, the factorial designs for 2, 3 and 4 experimental variables are shown.To continue the example with higher numbers, six variables would give 2 6 = 64 experiments, seven variables would render 2 7 = 128 experiments, etc.The levels of the factors are given by -(minus) for low level and + (plus) for high level.A zero-level is also included, a centre, in which all variables are set at their mid value.Three or four centre experiments should always be included in factorial designs, for the following reasons: The risk of missing non-linear relationships in the middle of the intervals is minimised, and Repetition allows for determination of confidence intervals.What -and + should correspond to for each variable is defined from what is assumed to be a reasonable variation to investigate.In this way the size of the experimental domain has been settled.For two and three variables the experimental domain and design can be illustrated in a simple way.For two variables the experiments will describe the corners in a quadrate (Fig. 1), while in a design with three variables they are the corners in a cube (Fig. 2).
The experiment in a design with two variables 6. Signs of interaction effects [5] The sign for the interaction effect between variable 1 and variable 2 is defined as the sign for the product of variable 1 and variable 2 (Table 2).The signs are obtained according to normal multiplication rules.By using these rules it is possible to construct sign columns for all the interactions in factorial designs.Example 1: A 'work-through' example with three variables This example illustrates how the sign tables are used to calculate the main effects and the interaction effects from a factorial design.The example is from an investigation of the influence from three experimental variables.

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Wide Spectra of Quality Control 24 The experiment in a design with three variables

Fractional factorial design
To investigate the effects of k variables in a full factorial design, 2k experiments are needed.Then, the main effects as well as all interaction effects can be estimated.To investigate seven experimental variables, 128 experiment will be needed; for 10 variables, 1024 experiments have to be performed; with 15 variables, 32,768 experiments will be necessary.It is obvious that the limit for the number of experiments it is possible to perform will easily be exceeded, when the number of variables increases.In most investigations it is reasonable to assume that the influence of the interactions of third order or higher are very small or negligible and can then be excluded from the polynomial model.This means that 128 experiments are too many to estimate the mean value, seven main effects and 21 second order interaction effects, all together 29 parameters.To achieve this, exactly 29 experiments are enough.On the following pages it is shown how the fractions (1/2, 1/4, 1/8, 1/16 . . .1/2 p) of a factorial design with 2 k-p experiments are defined, where k is the number of variables and p the size of the fraction.The size of the fraction will influence the possible number of effects to estimate and, of course, the number of experiments needed.If only the main effects are to be determined it is sufficient to perform only 4 experiments to investigate 3 variables, 8 experiments for 7 variables, 16 experiments for 15 variables, etc.This corresponds to the following response function: It is always possible to add experiments in order to separate and estimate interaction effects, if it is reasonable to assume that they influence the result.This corresponds to the following second order response function: In most cases, it is not necessary to investigate the interactions between all of the variables included from the beginning.In the first screening it is recommended to evaluate the result and estimate the main effects according to a linear model (if it is possible to calculate additional effects they should of course be estimated as well.).After this evaluation the variables that have the largest influence on the result are selected for new studies.Thus, a large number of experimental variables can be investigated without having to increase the number of experiments to the extreme.

Optimization
In this part, two different strategies for optimization will be introduced; simplex optimization and response surface methodology.An exact optimum can only be determined by response surface methodology, while the simplex method will encircle the optimum.simplex is a geometric figure with (k+1) corners where k is equal to the number of variables in a k-dimensional experimental domain.When the number of variables is equal to two the simplex is a triangle (Fig. 16.).
Var. 2 Var. 1 In this way, a new simplex is obtained.The co-ordinates (i.e., the experimental settings) for the new corner are calculated and the experiment is performed.When the yield is determined, the worst of the three corners is mirrored in the same way as earlier and another new simplex is obtained, etc.In this way, the optimization continues until the simplex has rotated and the optimum is encircled.A fully rotated simplex can be used to calculate a response surface.The type of design described by a rotated simplex is called a Doehlert design.

1 2 3 Fig. 3 .
Fig. 3.A simplex in two variablesSimplex optimization is a stepwise strategy.This means that the experiments are performed one by one.The exception is the starting simplex in which all experiments can be run in parallel.The principles for a simplex optimization are illustrated in Fig.17.To maximize the yield in a chemical synthesis, for example, the first step is to run k+1 experiments to obtain the starting simplex.The yield in each corner of the simplex is analyzed and the corner showing the least desirable result is mirrored through the geometrical midpoint of the other corners.In this way, a new simplex is obtained.The co-ordinates (i.e., the experimental settings) for the new corner are calculated and the experiment is performed.When the yield is determined, the worst of the three corners is mirrored in the same way as earlier and another new simplex is obtained, etc.In this way, the optimization continues until the simplex has rotated and the optimum is encircled.A fully rotated simplex can be used to calculate a response surface.The type of design described by a rotated simplex is called a Doehlert design.

Fig. 4 .
Fig. 4. Illustration of a simplex optimization with two variables The simplest polynomial model contains only linear terms and describes only the linear relationship between the experimental variables and the responses.In a linear model, the two variables x1 and x2 are expressed as: 012(,,, .) bbbe t cthat are to be determined.For the different models different types of experimental designs are needed.