In an age where data drives decisions, understanding the intricacies of the design of experiments is paramount.
Whether in business, engineering or even our daily lives, the ability to discern how different factors affect outcomes can lead to more informed decisions. The Design of Experiments (DOE) is a powerful tool for this purpose.
This article delves into the principles of DOE, introducing various types of experimental designs and illustrating their applications with concrete examples.
What is the Design Of Experiments (DOE)?
The Design of Experiments (DOE) was conceptualized to optimize experimental procedures and reduce their associated costs. In essence, it establishes a structure to understand the effects of independent variables (x) on a target response (y).
Design of experiments, or experimental design, aims to control an environment and vary certain factors (independent variables) to measure their impact on the desired response.
One of the notable proponents of DOE, Minitab, a company known for its robust statistical software, defines the design of an experiment as a mechanism that “[allows for] the simultaneous analysis of input variables’ effects on an output variable.” The process involves a set of trials where these input variables undergo intentional modifications. (Minitab, 2022)
When portrayed as a linear model, this relationship can be denoted as: y = ax + b, where :
 Y represents the response.
 a is the coefficient or slope.
 X is an independent variable.
 b is the intercept.
In the evercomplex landscape of business, various independent variables might affect our desired outcome. Considering the high costs associated with experimentation, it’s crucial to devise your experiments astutely, ensuring accurate conclusions from minimal trials.
Types of Design of Experiments
There are different types of experiment designs and each has its pros and cons. In this section, some plans will be presented and illustrated using examples.
Nested experiment plan :
Nested designs focus on specific scenarios where one independent variable is unique to another, termed “nested” factors.
Consider the following example:
A sports researcher studies the speed of a baseball as it leaves 4 different automatic pitchers (L). Each launcher has 3 different barrels (C) to expel the balls. The guns are unique and specific to each automatic launcher. They cannot, therefore, form a factorial plane (described a little later in the text).
This model does not contain interactions between variables. That is to say that the combination of 2 variables, here (L) and (C), cannot have an impact on the result. If the barrels were standard and interchangeable from one machine to another, we could have considered the interactions between the variables. We will come back to this in the section on factorial plans.
So the equation for this model would be: Yijk = μ + Li + Cj(i) + ε(kij)
Where “μ” represents the mean and “ε” the error
i = 1,2,3,4 (because there are 4 automatic launchers), j = 1,2,3 (3 guns per launcher) and k = 1,2,3
Factorial Plan :
Factorial designs explore the influence of every factor and their interactions. The strength of these designs is their thoroughness, but adding more factors can lead to a drastic increase in trials.
Each time a factor is added to the experiment, the number of trials increases exponentially (f^n, where f is the number of factors and n, is the number of levels).
Each variable can have several levels. The levels correspond to the number of “states” for which a variable will be tested. For example, we want to compare the time it takes two microwaves to cook a bag of popcorn with and without butter. One of the microwaves has a power of 1800 Watts, while the other has a power of 2200 Watts. The “power” and “type of popcorn” factors (with or without butter) therefore have 2 levels. We are therefore talking about plan 22, which gives 4 experiences.
The factorial plans also make it possible to check the effect of the interactions of variables on the answer. It is considered that there is an interaction between two or more variables when “the combined effect [of] the variable is greater than the effects of each of them taken separately” (Hicks & Turner, 1999).
Mathematically, a factorial plan, where A and B are the 2 factors at 2 levels, results in the following formula: Y= μ + Ai + Bj + ABij + εk(ij). AB represents the interaction between factors A and B represents the mean, and “ε” represents the error.
Fractional Design of Experiments :
Fractional designs are a more economical choice. By only performing a fraction of the full factorial design, we can still glean substantial insights. However, this requires making assumptions about which interactions might be negligible.
These plans of experiments make it possible to obtain practically as much information as a factorial plan (complete plan), but by carrying out only a fraction of this one. The number of trials then corresponds to f^{nk}.
To do this, the experiment must be divided into “blocks”, in which “it must be decided which effects can be confounded”. (Hicks & Turner, 1999).
That is, it is assumed that some factor interactions are negligible. In their book, Hick & Turner mentions 5 steps to determine the blocks of a 2^{f} plane (2level variables):
 Define the relation to be confused with I (identity)

Define factor levels (0 = Low level; 1 = High level)

ki represents the exponent of the variable (factor) of the relation defined in step 1. If the variable is not in the relation, its exponent is 0. We will give a value of 1 for all the xi.

Calculate the following linear expression for each combination: L = k1x1 + k2x2 + ⋯ + kfxf, mod 2 where mod 2 is the modulo 2, i.e. the “binary operation which associates with two natural integers the remainder [of a division ]. (Modulo, 2022). Example, 7 Mod 2 = 1, because 7 = 2×3 + 1. Following this logic, 8 mod 2 = 0, because 8 = 2×4 + 0.
 All combinations having the same value of L form a block.
Consider the following example, with a 2^{31} fractional plan where
Y = μ + A + B + C + AB + AC + BC + ABC + ε
Let’s define I = ABC, we get the following blocks:
Block 1 : (1), ab, ac et bc
Block 2 : a, b, c, abc
where (1) represents the combination where all the factors are at low levels (0).
Demonstration :
If I = ABC, then : I = ABC = A^{1}B^{1}C^{1}
 L(ABC) = 1×1 + 1×1 + 1×1 mod 2 = 3 mod 2 = 1
Now, let’s take each combination :
A = A^{1} = 1×1 mod 2 = 1 mod 2 = 1
B = B^{1 }= 1×1 mod 2 = 1 mod 2 = 1
C = C^{1} = 1×1 mod 2 = 1 mod 2 = 1
AB = A^{1}B^{1} = 1×1 + 1×1 mod 2 = 2 mod 2 = 0
AC = A^{1}C^{1} = 1×1 + 1×1 mod 2 = 2 mod 2 = 0
BC = B^{1}C^{1} = 1×1 + 1×1 mod 2 = 2 mod 2 = 0
(1) = A^{0}B^{0}C^{0} = 0x0 + 0x0 + 0x0 mod 2 = 0 mod 2 = 0
By grouping the combinations having an equivalent value of L, we obtain the 2 blocks mentioned above (Block 1 = (1), ab, ac, and bc / Block 2 = a, b, c, and abc).
The block containing the combination (1) is considered the main block. However, the study block will be chosen randomly to carry out the experiment.
The preceding calculations demonstrate the principle of the methodology presented by Hick & Turner.
However, there are tables where we find the I relations and the main blocks according to the factorial plane and the number of levels (f). Here is what these tables look like (example with 2^{f} planes):
f  Block(s)  Relation (I)  Principal Block 
2  2  I = AB  (1), ab 
3  2  I = ABC  (1), ab, ac, bc 
4  I = AB = AC = BC  (1), abc  
4  2  I = ABCD  (1), ab, bc, ac, abcd, cd, ad, bd 
4  I = ABC = BCD = AD  (1), bc, acd, abd  
8  I = AB = BC = CD  (1), abcd 
Other blocks can be generated from the main block. To do this, you must select an element other than that of the main block and determine the product of this element with the elements of the main block.
Example: The main block is as follows: (1), ab. So the second block will be: a, b
(1) * A^{1} = A^{1}B^{0} = A ou (a)  (1) * B^{1}= A^{0}B^{0} *B^{1} = B ou (b) 
AB * A = A^{1}B^{1} *A^{1} = A^{2}B^{1} = B (b)  AB * B =A^{1}B^{1} *B^{1} = A^{1} *B^{2} = A (a) 
* Lowercase letters are standard when talking about aliases (combination in blocks)
Taguchi Plan :
Finally, the last experimental plans presented are the Taguchi plans. They were developed starting in the 1950s by engineer and statistician Gemichi Taguchi.
The Taguchi plans aim to manufacture quality products, resistant to environmental fluctuations while minimizing the experiments required to verify which factors influence the desired response.
Within these planes, there are different orthogonal planes. The notation for these plans is as follows: L_{x }where x represents the number of trials performed to check the effect of y factors on a response.
To illustrate this concept, here is a chart with some Taguchi plane notations and their meanings:
Notation  Signification 
L_{4 }ou 2^{3}  4 attempts can be made to assess a maximum of 3 factors at 2 levels. 
L_{8 }ou 2^{7}  8 attempts can be made to evaluate a maximum of 7 factors at 2 levels. 
L_{9} ou 3^{4}  9 attempts can be made to evaluate a maximum of 3 factors at 3 levels. 
L_{16} ou 2^{15}  16 attempts can be made to evaluate a maximum of 15 factors at 2 levels. 
L_{27} ou 3^{13}  27 attempts can be made to evaluate a maximum of 13 factors at 3 levels. 
Taguchi’s designs of experiments use triangular tables and line graphs to determine where to position factors and their interactions. The graphs are predetermined according to the plans to be made.
For example, for an L8 plane, we find the triangular table and the choice of graphs below. The choice of the graph is made according to the interactions that we want to consider.
Triangular table L_{8} (2^{7}) :
Column  1  2  3  4  5  6  7 
(1)  3  2  5  4  7  6  
2  1  6  7  4  5  
(3)  7  6  5  4  
(4)  1  2  3  
(5)  3  2  
(6)  1  
(7) 
* The numbers in this table correspond to the numbers on the line graphs (below)
Linear Graph L_{8} (2^{7})
In the first graph, the one represented by a triangle, the numbers 3, 5, and 6 represent interactions. For example, 6 is the interaction between factors 2 and 4.
Following this same concept, the numbers 3, 5, and 7 represent the interactions in the second graph.
Finally, here is a brief example of a mathematical formula for an L8 plane, along with its triangular table and line graph.
Example :
We want to assess the impact of certain factors on the speed of a slap shot by a hockey player. The factors (at 2 levels) under consideration are: stick flex (A), blade curve (B), stick length (C), and stick weight (D).
The answer for this study is the speed of the puck as it enters the net. The place where the throw is made and the athlete is constant (factors controlled). So we get the following formula:
Y = μ + A(i) + B(j) + AB(ij) + C(k) + AC(ik) + D(l) + ε(m(ijkl))
Triangular table :
Column  A 1  B 2  AB 3  C 4  AC 5  6  D 7 
(1)  1  1  1  1  1  1  1 
cd  1  1  1  2  2  2  2 
bd  1  2  2  1  1  2  2 
bc  1  2  2  2  2  1  1 
ad  2  1  2  1  2  1  2 
ac  2  1  2  2  1  2  1 
ab  2  2  1  1  2  2  1 
abd  2  2  1  2  1  1  2 
* The 1 and 2 represent the levels of the variables
Linear Graph :
Note: The empty 6 adds to the error denoted ε. This increases the number of degrees of freedom (df) of error and thereby increases the accuracy of the experimental design results.
Conclusion:
Understanding and implementing the Design of Experiments is pivotal for anyone looking to make datadriven decisions.
Whether aiming for business optimization or scientific research, a wellcrafted experimental design can offer invaluable insights while conserving resources.
As we navigate an increasingly datacentric world, such tools empower us to operate more efficiently and make betterinformed choices.
Reference :
Abdulnour, Samir (2021). Impact de l’implantation de principe et d’outil du 4.0 et de l’agilité dans une PME québécoise – étude par simulation. Mémoire. TroisRivières, Université du Québec à TroisRivières, 104 p
Abdulnour, S., Baril, C., Abdulnour, G., & Gamache, S. (2022). Implementation of Industry 4.0 Principles and Tools: Simulation and Case Study in a Manufacturing SME. Sustainability, 14(10), 6336.
Minitab (2022). https://support.minitab.com/frfr/minitab/18/gettingstarted/designinganexperiment/
Modulo (opération). (2022, mai 18). Wikipédia, l’encyclopédie libre. Page consultée le 26 mai 2022 à partir de https://fr.wikipedia.org/wiki/Modulo_(op%C3%A9ration)
Plan factoriel. (2017, novembre 19). Wikipédia, l’encyclopédie libre. Page consultée le 17 Mai 2022 à partir de http://fr.wikipedia.org/w/index.php?title=Plan_factoriel&oldid=142757878.
R. Hicks, C. & V. Turner, K. (1999). Fundamental concepts in the design of experiments, Oxford University Press, 5^{th}.