Scientific insights from disciplines such as Neurosciences reveal how blurred is the way we humans analyze, conclude and make our decisions(compare https://insights.athena.edu/blog/mathematical-similarities-between-neuroeconomics-and-machine-learning/ ). But, as we perfectly know, our approaches still work. Otherwise we would not have been able to send a man to the moon, land the “Curiosity” rover on the surface of Mars, or – more recently – understand the mathematics of black holes. However, our neuro systems conceive most of the phenomena and objects around us not in an explicit, distinct way but rather ambiguously. For example, if I were told that pursuing a Doctorate degree is very time consuming, whereas undertaking a Master degree is just moderate time consuming and studying for a Bachelor Degree is not very time consuming; what would that exactly mean? And even more interesting for every Business Administration student, how can we possibly find a way to model such fuzzy statements, for instance, for decision making in Management where the truth is not always black and white? In the light of a paper I am currently preparing, I want to present you a mathematical approach which enables us to describe and model those linguistic, rather imprecise statements. The methodology is called “Fuzzy Logic”.
Classic Set Theory
Since you are reading this article, I can infer that you are interested in pursuing graduate studies, that is, you are probably looking for a suitable MBA or even a PhD or DBA program and hence – at least to some extent – I can assume that you are familiar with set theory from your undergraduate studies. Now, as you will remember, in set theory we can decide whether or not a single object, such as a number, belongs to a set or not, simply by examining its properties and then comparing these properties with the set’s characteristics. For example, if A is the set of all-natural numbers between 3 and 11 then we can easily conclude that 4 is a member of A, or in mathematical notation, 4 E A. And the statement 4 E A is then 100% true. This is called a crisp set theory model and we can model a related membership function in the following binary (black and white) way:
f(4)=1, because 4 is a member of A. Let us now take the initial example of DBA, Master and Bachelor studies. Can we model whether an activity is very time consuming, moderate time consuming or not very time consuming with crisp set theory?
Let us assume that we have taken a survey where we asked a sufficient number of students about what they consider as very time consuming, moderate time consuming and not very time consuming. And let us say these are the ranges they came up with:
X: Very time consuming: 11 to 50 hours a week
Y: Moderate time consuming: 7 to 15 hours a week
Z: Not very time consuming: 2 to 10 hours a week
As you can easily see, a simple crisp set description for a membership function such as the one above is not very helpful in this case. For example, if a student studies, let’s say 12 hours a week, are her studies then to be considered as being moderate time consuming or very time consuming? You might argue that since 12 is closer to Y`s than to X`s median, 12 hours should be considered as being moderate rather than very time consuming. But what about 15.1 hours a week? Can you then argue “well last year my studies were moderate time consuming because I spent 15 hours a week studying. But this year is completely different, my studies are very time consuming, because on average I have studied 15.1 hours a week”? Well, you got the idea.
Fuzzy logic enables us to model such problems. Fuzzy logic was “invented” by Lotfi A. Zadeh. In 1965, a first few papers in print were published about sets with a graded membership relation (Gottwald, S., 2008).
The idea of fuzzy logic is to assign a weight of how much an element belongs to a fuzzy set by a number which we consider as the “degree of being a member” with a range covering the interval (0,1), where 1.0 means absolute truth and 0.0 means absolute falseness.
There are plenty of “ready-to-go” membership functions which can be deployed for a variety of applications. An easy to grasp one is the Triangular Function which is based on the intercept theorem, also known as Thales’s theorem. The Triangular Function is defined by a lower limit a, an upper limit b, and a value m, where a < m < b:
One can comprehend the beauty of the approach by comparing its graph with the “black and white” (binary) one: As per the binary relation, studying 3 hours is not very time consuming. The same applies for any value between 2 and 10: It is always 100% “not very time consuming”. But as per the triangular membership function, studying 3 hours a week is about 25% ”not very time consuming”.
Binary membership function for Z: Not very time consuming: 2 to 10 hours a week.
Triangular membership function for Z: Not very time consuming: 2 to 10 hours a week
The real power of this approach is revealed if one combines different fuzzy membership functions. For this purpose, let us expand our definition of “study time consumption”. Let’s pretend there are two items (=input variables) to be considered to decide whether studying is not very time consuming, moderate time consuming or very time consuming plus one output variable. The input variables are now the time spent for studying, and the average income per hour a student would earn if she works instead of studying. The output variable shall be the “time consumption of the studies”.
A: time spent for studying (in hours), where
A=little if 2<=u(x)<=10
A=moderate if 7<=u(x)<=15
A=much if 11<=u(x)<=50
B: income per hour (in $), where (for example)
B=small if 5<=u(x)<=10
B=average if 9<=u(x)<=15
B=high if 13<=u(x)<=20
(notice the imprecise (“fuzzy”) but common linguistic values of the input variables A and B)
D: study (effort)
Possible outcome values of D shall be “not very time consuming”, “moderate time consuming” and “very time consuming”.
Let us also assume that we asked some experienced students (in fuzzy logic, they are usually called experts) to set up rules (called “rules of inference”) for us to decide which combination of the input variables lead to which values of the output variable. The experts came up with the following rules.
By combining the membership functions using the AND and OR operators appropriate algorithms now can determine a specific centre of weight of the area under the combined curve. The x position of that centre is then the final output. It goes without words that a detailed description of the algorithms goes beyond the scope of this article. However, common programming languages like python or MATLAB offer ready-to-go solutions.
In the second part of this article I will show you how to actually implement our example with MATLAB. Stay tuned…
Gottwald, S., (2008), “Fundations of a set theory for fuzzy sets – 40 Years of development”, https://www.researchgate.net/publication/4197625_Foundations_of_a_set_theory_for_fuzzy_sets_40_Years_of_development