 # Fuzzy set theory and fuzzy logic in management

Get a clearer view of uncertain, or fuzzy, sets and some of their latest applications in the field of management from Eötvös Loránd University’s Tamás Jónás

Nowadays, fuzzy sets and fuzzy logic have a wide range of applications in many areas of science including engineering, economics, management, and the business sciences.

This article offers an insight into the research results achieved over the past five years by the Faculty of Economics, Eötvös Loránd University, in fields connected with fuzzy set theory, fuzzy logic, and their applications in management.

Basic notions

Let us consider the statement: ‘Our company has a high operating profit rate.’ We do not know exactly what this statement means, but in our daily life, we tend not to see any problem with using this kind of statements. This is because we have a certain imagination of the meaning of the above statement, even though it is ambiguous and imprecise.

In classical set theory, the set of high operating profit rates can be exactly defined as an interval on the real line. For example, if the operating profit rate is between 10% and 30%, then we may consider it as high operating profit rate.

Now, suppose that an operating profit rate, x, is just a bit less than 10%. In this case, noting the previous crisp definition of the set of high operating profit rates, x is not an element of this set. However, we would describe this situation using statements like: ‘The operating profit rate, x, is almost high’ or ‘the operating profit rate, x, is close to a high operating profit rate’. Our human language allows us to express our uncertainty regarding this situation in a more sophisticated way than that of using the concept of crisp set. Based on this line of thinking, we may conclude that the classical set theory is not suitable for describing the uncertainty associated with vague situations.

The concept of fuzzy set was introduced by University of California, Berkeley Professor, Lotfi Zadeh, in 1965 as an extension of the classical notion of set. A fuzzy set is given by its membership function, which is a mapping from a given universe (set) to the closed unit interval.

The value of the membership function at x is the grade of membership of x in the fuzzy set. It should be emphasised that every element in the universe has its membership grade in the fuzzy set in question, and this grade is an element of the unit interval [0,1].

Using our previous example, we can define the fuzzy set of high operating profit rates such that the membership grade of 9% operating profit rate to this fuzzy set is 0.3. That is, this operating profit rate has a relatively low membership grade in the fuzzy set of high operating profit rates, but this membership grade is not zero.

We can see that this set has no crisp boundaries, and that its boundaries are instead rather ‘fuzzy’. This explains why these sets are called fuzzy sets. It should be added that the membership grade of x, which tells us how much x belongs to the fuzzy set on the scale [0,1], may also be viewed as a measurement of truth of the statement that ‘x is an element of the set under study’. Here, the values zero and one correspond to the ‘fully’ false and ‘fully’ true logical values, respectively. That is, the value of the membership grade can be interpreted as a fuzzy logical (continuous logical) value.

Evaluations using fuzzy numbers

A special class of fuzzy sets, called the fuzzy number, may be treated as an extension of the notion of number. Namely, a fuzzy number can be used to express the uncertainty related to a quantity. Moreover, using fuzzy numbers, traditional Likert scale-based evaluations can be extended to fuzzy rating scale-based evaluations that find many applications in the business sciences. For example, in relation to evaluating service features in healthcare, or the performance of university lecturers and supervisors.

Ranking fuzzy numbers and multicriteria decision-making

The ranking of fuzzy numbers plays an important role in multicriteria decision-making, and as such, it has various applications in management as well.

Fuzzy logical operations

In many situations, we need to modify a fuzzy logical value or connect multiple fuzzy logical values to obtain a new one. These can be done using continuous-valued logical operators that may be viewed as extensions of the negation, conjunction, and disjunction operations of classical logic. You can see an example of our results connected with this area here.

Monotone (fuzzy) measures in decision-making and behavioural economics

The monotone, but not necessarily additive measures (set functions), which are also known as fuzzy measures, can be used to model various phenomena in decision-making and in behavioural economics. Our contributions to these fields include, for example, a methodology that can be used to generate parametric probability weighting functions.

Demand forecasting using fuzzy logic

Forecasting the demand for products or services is a crucial activity for all enterprises. In the past few years, we developed some soft computational methods that can be used in demand forecasting effectively, an example of which can be seen in a 2018 paper in the International Journal of Production Economics.

Tamás Jónás is an Associate Professor at the Faculty of Economics, Eötvös Loránd University (ELTE) in Budapest, Hungary.

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