Adaptation of Fuzzy Systems Based on Ordered Fuzzy Numbers: A Review of Applications and Development Prospects

Abstract

This paper presents a comprehensive overview of the adaptation of fuzzy systems based on Ordered Fuzzy Numbers (OFNs), an extension of classical fuzzy set theory that allows for more accurate modeling of uncertainty and variability across diverse domains. Key adaptation techniques—including genetic algorithms, evolutionary programming, learning algorithms, reinforcement learning, and online adaptation—are systematically analyzed and compared in terms of their strengths, limitations, and application areas. The analysis reveals that, despite the considerable potential of OFN-based systems in fields such as engineering and the social sciences, current adaptation methods encounter challenges related to computational complexity, scalability, and real-time implementation. This work aims to provide a comprehensive overview of the state of the art in the field and inspire further research on OFN applications in various areas of science and technology.

1. Introduction

Ordered Fuzzy Numbers (OFNs) are an important step in the development of fuzzy systems, as they enhance their possibilities. OFNs make it possible to solve the limitations of classical fuzzy numbers while introducing a dynamic orientation to model complex phenomena. OFNs have redefined fuzzy arithmetic by representing numbers as ordered pairs of continuous functions, enabling precise algebraic operations and eliminating the loss of accuracy associated with traditional approaches [1,2]. Such solutions have solved problems encountered in fuzzy arithmetic, such as improper subtraction and problematic division. In addition, orientation has been introduced as a key feature to capture dynamic trends in the data, such as rising or falling market values or time-dependent processes [1,3,4].

Over the past two decades, OFNs have evolved from theoretical constructs to practical tools that can be used in many disciplines. Initial applications focused on financial modeling, such as evaluating project profitability using OFNs [4]. Later research expanded into multi-criteria decision-making (MCDM), where OFNs improved methods such as fuzzy SAW (Simple Additive Weighting) and TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) by incorporating linguistic evaluations and dynamic criteria [5,6,7]. Over the past 2 years, there have been significant advances in OFNs. Most notably, OFNs have evolved considerably, enabling integration with neural networks in energy-efficient quality-of-life technologies [8]; IoT (Internet of Things) systems for real-time optimization of energy consumption [9]; real-time order book analysis in stock markets [3];and adaptive supplier evaluation systems in supply chain management [10]. The aforementioned applications highlight OFNs’ ability to handle uncertainty, model temporal changes, and correct computational resilience in high-frequency environments.

Theoretical improvements—such as the revision of OFN algebra [2]—have further strengthened the mathematical basis by providing consistency. Today, emerging fields such as cybersecurity [11] and increasingly important healthcare sector [12] demonstrate the growing importance of OFNs. Current research trends are moving toward hybrid models that combine OFNs with deep learning [10], suggesting that there is still untapped potential in systems based on AI and real-time adaptive control.

Regarding the methods developed and used so far for traditional fuzzy numbers, the specifics of OFNs mean that classical approaches can serve as inspiration. However, the extent of the necessary modifications may be significant enough to warrant a separate set of methods dedicated to OFNs.

This review is intended to analyze the adaptation of fuzzy systems based on OFNs. First, the definition of OFNs is presented, along with examples and arithmetic properties. Then, the role of OFNs in various application domains—including finance, MCDM, engineering, and AI—is then analyzed. The subsequent sections address key challenges, such as interpretability in complex systems, and outline development prospects for OFN-based fuzzy systems. Special attention is given to adaptation methods, including genetic algorithms, evolutionary programming, learning algorithms, reinforcement learning, and online adaptation. The Discussion section highlights the limitations of the current review and identifies key directions for future research. The paper concludes with a summary of the main findings and the relevant recommendations. This work aims to provide a comprehensive path for theorists and practitioners using OFNs in fuzzy systems.

2. Ordered Fuzzy Numbers

2.1. Definition of OFNs and Arithmetic Properties

Fuzzy set theory, introduced by Zadeh (1965) [13], has revolutionized the approach to modeling uncertainty and imprecision in science [14]. It has made it possible to model phenomena that are difficult to describe using classical mathematical methods [14]. However, the development of applications based on fuzzy set theory has revealed some limitations of the classical approach. The solution to issues with classical fuzzy set theory is the concept of OFNs. This model, proposed by Kosiński, Prokopowicz, and Ślęzak, was created to cope with the limitations of classical fuzzy set theory [1]. OFNs are a natural extension of classical fuzzy numbers, preserving their advantages while introducing new possibilities [1,2].

Definition 1. 

OFNs are defined as an ordered pair of continuous real functions and can be expressed as follows:

$$\mathit{A}=\left(\mathit{f},\mathit{g}\right),\mathit{f},\mathit{g}:\left[\mathrm{0,1}\right]\to \mathit{R},$$

where

• f (increasing part) represents the up-part of the fuzzy number;
• g (decreasing part) represents the down-part of the fuzzy number [11,15].

Figure 1 shows a graphical interpretation of Ordered Fuzzy Numbers with upper and lower parts. We can distinguish the three subfigures as follows:

(a)

shows an Ordered Fuzzy Number, where the functions f and g represent, respectively, the upper (UP) and lower (DOWN) parts of the fuzzy number. The axes denote the variables x and y, and the lines represent the ranges of these functions.

(b)

represents fuzzy numbers in the classical sense, where the functions f⁻¹ and g⁻¹ invert the original relationships. An additional interval (marked with a horizontal line) illustrates the difference between the ordered representation and the classical approach.

(c)

Simplistically denotes the order of the inverse functions with bold arrows, emphasizing that we are dealing with an ordered pair of functions.

The whole figure illustrates the idea of representing fuzzy numbers with functions f and g and their inverses, and introduces auxiliary signs, such as the orientation of the arrow in (c) [15].

Definition 2. 

Let A = (f_A, g_A), B = (f_B, g_B) and C = (f_C, g_C) be OFNs. Actions on these objects, such as sum C = A + B, difference C = A − B, product C = A · B, and quotient C = A/B, are defined by the following formula:

$${\mathit{f}}_{\mathit{C}}\left(\mathit{y}\right)={\mathit{f}}_{\mathit{A}}\left(\mathit{y}\right)\star {\mathit{f}}_{\mathit{B}}\left(\mathit{y}\right)\wedge {\mathit{g}}_{\mathit{C}}\left(\mathit{y}\right)={\mathit{g}}_{\mathit{A}}\left(\mathit{y}\right)\star {\mathit{g}}_{\mathit{B}}\left(\mathit{y}\right)$$

where the symbol ⋆ stands for“+”, “−”, “−” and “/” operations, respectively. Note, however, that the quotient A/B is determined only if zero does not belong to the intervals defined by the functions f_B(y) and g_B(y).

Subtraction of the fuzzy number B reduces to the addition of its opposite, that is, the number (−1) · B. If for a fuzzy number A= (f, g), we define its complement as $\overline{\mathit{A}}=(-\mathit{g},-\mathit{f})$, their sum results in a fuzzy number zero, as follows:

$$0=(\mathit{f}-\mathit{g},-\left(\mathit{f}-\mathit{g}\right))$$

which corresponds to the classical definition from the calculus of fuzzy numbers [15].

The above-mentioned operations retain algebraic properties such as connectivity and alternation between addition and multiplication [16]. In the above-mentioned way, many operations suitable for pairs of functions can be defined. The Fuzzy Calculator was developed as a computational tool by Mr. Roman Koleśnik [17]. It enables the simplified use of all mathematical objects described as OFNs in future applications.

Example operations on OFNs are presented in Figure 2, Figure 3, Figure 4 and Figure 5 [11].

Definition 3. 

Let A = (f_A, g_A), B = (f_B, g_B), and C = (f_C, g_C) be OFNs, where C is the result of an operation on A and B.

The maximum of A and B, denoted as $\mathit{C}=\mathit{A}\bigvee \mathit{B}$, is computed by taking the maximum value of their first functions and the maximum value of their second functions at each point y, as expressed in the following equation:

$${\mathit{f}}_{\mathit{c}}\left(\mathit{y}\right)=\mathbf{m}\mathbf{a}\mathbf{x}\{{\mathit{f}}_{\mathit{A}}\left(\mathit{y}\right),{\mathit{f}}_{\mathit{B}}\left(\mathit{y}\right)\}$$

$${\mathit{f}}_{\mathit{c}}\left(\mathit{y}\right)=\mathbf{m}\mathbf{a}\mathbf{x}\{{\mathit{f}}_{\mathit{A}}\left(\mathit{y}\right),{\mathit{f}}_{\mathit{B}}\left(\mathit{y}\right)\}$$

The minimum of A and B, denoted as $\mathit{C}=\mathit{A}\wedge \mathit{B}$, is computed by taking the minimum value of their first functions and the minimum value of their second functions at each point y, as follows:

$${\mathit{f}}_{\mathit{c}}\left(\mathit{y}\right)=\mathbf{m}\mathbf{i}\mathbf{n}\left\{{\mathit{f}}_{\mathit{A}}\left(\mathit{y}\right),{\mathit{f}}_{\mathit{B}}\left(\mathit{y}\right)\right\}$$

$${\mathit{g}}_{\mathit{c}}\left(\mathit{y}\right)=\mathbf{m}\mathbf{i}\mathbf{n}\{{\mathit{g}}_{\mathit{A}}\left(\mathit{y}\right),{\mathit{g}}_{\mathit{B}}\left(\mathit{y}\right)\}$$

The “max” operation selects the highest values from the corresponding functions of A and B, while the “min” operation selects the lowest values [15].

2.2. The Role of OFNs

Adaptation of fuzzy systems based on OFNs is a growing area of research that combines the strengths of OFNs with adaptive methods to enhance the performance and flexibility of fuzzy systems. One of the most promising areas of research on OFN-based adaptive systems is the integration of neural learning techniques. Recent studies have shown that combining neural networks with OFNs can lead to significant improvements in performance and adaptability [8]. Apiecionek et al. (2023) [8] proposed the idea of using OFN arithmetic to improve the quality of daily human life through better resource management. The approach involved the fuzzification of input data to the network and the defuzzification of output data from the network, followed by the development of algorithms for training networks that operate using OFN arithmetic. The authors presented that their OFN-based neural network achieved a computational error of 0.00036679%, significantly outperforming conventional deep neural networks, which showed a computational error of 0.04220253% in similar energy management applications [8]. Based on this study, it can be concluded that the integration of OFNs with neural networks offers many advantages, which are particularly important for applications in smart home systems, data transmission control, or server management in cloud systems [8]. Another important area of research is the development of adaptive fuzzy control techniques, which indicate the potential to increase the performance and resistance of control systems. Publications by Yang and Zhang (2024) [18], Sun et al. (2024) [19], and Mirzajani et al. (2019) [20] present various approaches to adaptive fuzzy control for fractional-order systems, demonstrating improvements in performance and resistance in the presence of uncertainty and nonlinearity. However, despite these advances, none of these works considered the application of OFNs in the context of adaptive fuzzy control for fractional-order systems. This represents an important gap in the current state of research, as OFNs offer unique capabilities for modeling uncertainty and directionality that could significantly improve the performance and robustness of control systems [18,19,20]. Adaptive techniques based on OFNs have also shown promise in time series forecasting, particularly in financial applications [3]. OFNs are well suited for modeling and predicting dynamic market behavior. Marszałek and Burczyński (2024) [21] proposed using OFNs to encapsulate the dynamic behavior of the LOB (Limit Order Book)—the central element of markets that matches buyers and sellers in the financial market [22]—thereby increasing robustness to data perturbations, such as removing or addition of irrelevant orders [3]. The proposed representation of LOB data has shown stable out-of-sample prediction accuracy, even when subjected to data perturbations [3]. While traditional fuzzy time series models have found widespread use in a variety of forecasting tasks [23,24], the integration of OFNs introduces significant benefits. The directional nature of OFNs enables a more accurate representation of trends, which is crucial in financial forecasting. In particular, in the analysis of financial markets, the direction of price changes has often played as important a role as the value of prices, and the use of OFNs makes it possible to better capture these correlations [21].

Key features that make OFNs valuable for fuzzy system adaptation include directionality [11], improved arithmetic [11], lower complexity [10], and increased modeling power [11]. OFNs’ directionality allows for modeling upward and downward trends, providing richer information than classical fuzzy numbers. This makes it possible to capture changes in fuzzy systems more precisely [11]. On the other hand, the improved arithmetic of OFNs, based on correctly defined arithmetic operations [11], enables more accurate calculations in fuzzy systems, paying attention to small differences between similar phenomena [25]. Czerniak et al. (2013) [26] claimed that operations performed using OFNs were less complicated, and their results were more accurate in most cases. They also noted that OFNs offered better precision in calculations, especially for small differences [26]. In turn, the lower complexity can be attributed to the fact that the use of OFNs can lead to simpler rule bases compared to classical fuzzy systems, which translates into better interpretability of models [10]. Kosinski et al. (2003) [1], who introduced the concept of Ordered Fuzzy Numbers, compared classical fuzzy numbers with OFNs, showing that a more flexible representation of fuzzy concepts could lead to simpler models [1]. Rudnik et al. (2021) [27] proposed a new approach to building more transparent knowledge-based systems by generating interpretable fuzzy rules [27]. Two years later, Rudnik and Chwastyk (2023) [25] developed the concept of an inference method for IF-THEN rules with OFNs that reduces the complexity of the knowledge base compared to classical fuzzy systems and enables more efficient modeling of dynamic changes in the inference process [25]. In turn, the increased modeling power derives from the additional information stored in OFNs, which allows for a more accurate representation of complex phenomena and increases the accuracy of fuzzy systems, from financial systems to life-enhancing technologies [3,8,11].

2.3. Challanges

Adapting fuzzy systems using OFNs presents several challenges that researchers must overcome.

Complexity in Defining and Interpreting OFNs

Determining the definition and proper interpretation of OFNs is very important [28]. To properly address this, it is necessary to outline how basic concepts, assumptions, and mathematical operations, which can often be complex, are defined. Complexity is increased by introducing specific parameters for OFNs, such as slope or order [11].

Developing Robust Arithmetic Operations

Implementing arithmetic operations for OFNs requires careful consideration to ensure consistency and reliability [4,29]. This includes defining operations that preserve OFN properties and developing methods to efficiently process directional information [2,11,15].

Integration with Fuzzy Control Systems

Using OFNs together with fuzzy control systems requires the development of new information processing methods. Using the arithmetic properties of OFNs can improve fuzzy rule adaptation; however, such integration requires innovative approaches and thorough testing [30,31].

Application in Multi-Criteria Decision-Making (MCDM)

The application of OFNs in MCDM involves representing various criteria, including those described linguistically. Developing methods such as fuzzy SAW and fuzzy TOPSIS based on OFNs [5,6,7] can be complex and requires methodological advances to deal with the inherent uncertainty and subjectivity in decision-making processes [10,32,33].

Handling Uncertainty and Subjectivity

One of the main objectives for using OFNs is the management of uncertainty and data subjectivity. However, effectively capturing and processing this uncertainty within OFNs remains a major challenge, requiring ongoing research and development [3,10,21].

Dealing with these challenges is crucial to advancing the use of OFNs in fuzzy systems and increasing their effectiveness in dealing with complex, uncertain information.

2.4. Development Prospects

In accordance with the current trends in the literature and due to the multidisciplinary nature of OFN research, the development prospects of the field can be clearly divided into two main areas—theoretical developments and practical implementations. With this division, both the fundamental research directions and the potential applications of OFNs in various scientific and economic fields can be better understood.

2.4.1. Theoretical Developments

The adaptation of fuzzy systems based on OFNs involves significant opportunities for development in various fields [10,21,34,35,36,37]. Computational intelligence and machine learning (ML) are constantly evolving. As a result, the integration of OFNs with fuzzy logic-based decision-making processes is expected to enhance precision, interpretability, and robustness in uncertain environments [3,38,39,40]. Future research is expected to focus on aspects such as improving algebraic structures, hybridizing with other theories, advancing fuzzy techniques, or formalizing learning mechanisms [41,42,43,44].

2.4.2. Practical Implementations

The practical impact of ordered fuzzy systems based on OFNs is expected to gain importance in a variety of economic sectors and scientific fields. AI, in combination with expert systems enriched through OFN-based fuzzy logic, can improve decision-making in applications such as robotics [8,45]. One particularly important discipline is medicine, where the incorporation of OFNs into diagnostic systems can enable better modeling of, for example, patient symptoms or uncertainties in medical data, leading to more accurate diagnoses and improved clinical support. By using fuzzy logic based on OFNs, it is possible to account for subjective assessments by doctors, variability in symptoms, and incomplete or contradictory information, resulting in more accurate clinical prognoses. Such an approach can be applied in medical decision support systems, the analysis of diagnostic images (e.g., X-ray, MRI, CT), as well as the real-time monitoring of patients’ conditions, allowing for earlier detection of health risks and faster response [12,46]. Finance and risk assessments have already been discussed several times in this article, but they are worth emphasizing again here. The financial sector can benefit greatly from OFN-based models for improved risk evaluation and more accurate forecasting of market trends under uncertain conditions [3,21,22,29]. While in finance, OFNs are typically used for modeling market uncertainties and assessing investment risks, their potential in the industrial sector is equally promising [4,10,47]. In project environments with high data volatility, such as critical infrastructure construction management or real-time software development, it is crucial to take uncertainty into account when estimating the progress of work [48]. Another important industry is autonomous systems and IoT. Incorporating OFNs into autonomous systems—such as self-driving vehicles and smart IoT devices—can contribute to more reliable decision-making under uncertainty. By accurately modeling ambiguous and dynamic data, OFNs can improve the ability of these systems to adapt to changing environments, anticipate potential risks, and optimize control processes. In particular, in autonomous vehicles, this can result in better object recognition, more precise responses to changing road conditions, and increased traffic management efficiency [8,49,50]. An equally important area of use for OFNs is the analysis of weather data, enabling more effective prediction of climate change and forecasting of extreme weather events, which translates into more effective risk management and planning across a range of sector [51].

2.5. Adaptation Methods

Methods for modifying knowledge systems based on fuzzy sets include techniques for adjusting or updating fuzzy knowledge and parameters in such systems. These are used to improve and optimize the performance of fuzzy systems over time or in response to changes in the environment [34].

Adaptation is the process of adjusting a knowledge system to better cope with dynamic changes in the environment or input data [52].To better understand advanced adaptation methods with systems based on OFNs, this section explores the use of genetic algorithms, evolutionary programming, learning algorithms, reinforcement learning, and online adaptation. These techniques enhance the flexibility and performance of OFN-based systems in dynamic and uncertain environments.

2.5.1. Genetic Algorithms

Genetic algorithms (GAs) are optimization methods based on natural evolution. In these algorithms, populations of fuzzy systems are generated with different sets of fuzzy rules and parameters. Through selection, crossover, and mutation operations, the algorithm iteratively improves the population and favors individuals with better performance.

Selection is a key determinant of the efficiency of GAs, directly influencing the rate of convergence and genetic diversity of populations. Its basic principle is based on a probability proportional to the value of the adaptation function. The simplest implementation of this principle is the so-called roulette wheel selection [53,54,55]. This method assumes that the probability of selection is proportional to the fitness of the individual. It can be described as follows. Let us consider N individuals, each characterized by their fitness w_i > 0 (i = 1, 2, …, N). The selection probability of the i-th individual is thus given as follows [53]:

$$p_{i }={\displaystyle \frac{w_i}{{\sum }_{i=1}^N{w}_i}}(i=1, 2, \dots , N).$$

This approach has been extensively described in the works of Goldberg and Mitchell [54,55], which are considered foundational in the literature on genetic algorithms [54,55,56].

GAs may adapt fuzzy systems to achieve optimal or near-optimal solutions [57]. A GA begins with a large population of potential solutions and, through the use of recombination (also known as crossover) and mutation, achieves a solution that is better than any previous solution at the time of the genetic analysis [58]. The process of crossover allows genetic information to be exchanged between the two parents, resulting in a new generation. The idea behind crossover is that a new chromosome can be superior to both parents if it takes the best traits from each parent [56]. In the single-point crossover scheme, a random point k is selected for two parents represented by strings of length L (with $1 \le k \le L-1$). Two new individuals are then created by combining fragments of the parents’ chromosomes, where x = (x₁, x₂, …, x_L) and y = (y₁, y₂, …, y_L) represent the parents, as follows:

$$z_1=(x_1, x_2, \dots , x_k, y_{k+1}, y_{k+2}, \dots , y_L),$$

$$z_2=(y_1, y_2, \dots , y_k, x_{k+1}, x_{k+2}, \dots , x_L).$$

Other types of crossovers are also in use, including the two-point crossover, multi-point crossover, and uniform crossover [54]. Examples of crossover operators have beenpreviously described in studies by Holland [59] and Goldberg [54].

Mutation is one of the key genetic operators. It is widely useddue to its simplicity and ease of implementation. In binary representations, the mutation mechanism involves changing the value of a single gene in a chromosome (from 0 to 1 or from 1 to 0). If we assume that x_i is the value of the gene in question before the mutation (0 or 1) and p_m is the probability of the mutation occurring, we define the value after the mutation operation x′_i as follows [54,57]:

$$x_i^{\prime }=\left\{\begin{array}{c}1-x_i, with probability p_m\\ x_i, with probability 1-p_m\end{array}\right.$$

The mutation model is a key tool to ensure population diversification, which is crucial in adaptive systems modeling, as it allows the algorithm to adapt to changing environmental conditions [59].

To further explain the general structure of a GA, a classical GA consists of the following steps: starting the process, initializing the population, evaluating the adaptation (fitness) of individuals, selecting the best individuals, applying genetic operators (such as crossover and mutation), and generating a new population [57].

Optimizing the parameters of a fuzzy system to find the best possible settings is one example of the application of GAs [60]. An example of such an application is the use of GA to optimize the performance of real-time multiprocessor systems, as described in the 2025 study. Hassan et al. [61] proposed a new approach based on GAs for scheduling tasks in multiprocessor systems, comparing their effectiveness with fuzzy set-based methods such as the Evolutionary Fuzzy-Based Scheduling Algorithm (EFSBA) [61]. The results indicate that the Optimized Performance Based Genetic Algorithm (OPBGA) not only provided zero task over deadline but also achieved the lowest average response and task completion times, even under high system load, highlighting its advantage over traditional methods. EFSBA also performed well, especially in low and moderate load scenarios. However, it should be noted that the high performance was achieved under the assumption of processor and task homogeneity and in a MATLAB simulation environment, which may limit the direct applicability of the results in real, more complex systems [61]. Another application involves adapting the parameters of the membership function in fuzzy systems to achieve optimal performance in a given situation [62]. In this approach, each subpopulation of parameters is processed in parallel, significantly reducing optimization time and enabling scalability in cloud environments. Research has shown that a heterogeneous population initialization strategy, combined with Particle Swarm Optimization (PSO) and GAs, can accelerate the search for optimal settings without sacrificing solution quality, which is particularly important in industrial applications such as autonomous vehicle control. In addition, the open and flexible architecture of this approach allows the system to be easily extended to accomodate new problems and metaheuristics, making it attractive for both small- and medium-sized enterprises [62]. GAs can advance both membership functions and rule sets of fuzzy systems, resulting in a better representation of uncertainty in complex datasets [63]. As a result, fuzzy systems designed using GAs demonstrate greater flexibility, better generalizability, and higher efficiency in reproducing the uncertainty and complexity of real-world problems [63].

2.5.2. Evolutionary Programming

Evolutionary programming extends the rules of GA and is a stochastic optimization method that uses a population of individuals to search for an optimal solution. It involves random variation, selection, and the survival of the best-adapted individuals. One of its applications is the adaptation of fuzzy rules and parameters in knowledge systems based on fitness evaluations, which improves system performance over time [64,65,66].

In evolutionary programming, an individual is represented by a vector of decision variables <x₁, x₂, …, x_k> and the corresponding strategy parameters, i.e., the $\sigma$ mutation steps. To allow for the adaptation of mutation intensity during evolution, a so-called self-adaptive mutation is used. The parameter $\sigma$ is mutated during each time step by multiplying it by a term $e^{\Gamma }$ with $\mathsf{\Gamma }$, a random variable drawn each time from a normal distribution with mean 0 and standard deviation $\tau$. Since $N=(0,\mathsf{\tau })=\mathsf{\tau }·\mathrm{N}(0,1)$, the mutation mechanism is therefore defined by the following formulas [67]:

$$\sigma =\sigma ·e^{\mathsf{\tau }·\mathrm{N}(0,1)},$$

$$x_i^{\prime }=x_i+\sigma \prime ·N_i(0,1).$$

where N (0, 1) denotes a draw from the standard normal distribution, while N_i (0, 1) denotes a separate draw from the standard normal distribution for each variable i. The proportionality constant $\mathsf{\tau }$ is an external parameter set by the user. It is usually inversely proportional to the square root of the problem size. The parameter $\mathsf{\tau }$ functions as an indicator regulating the rate of adaptation of the system, playing a role analogous to the learning rate coefficient used in neural network architecture. It determines the intensity with which the algorithm assimilates and integrates new information during evolutionary adaptation to the computational environment [67]. After the mutation operation, individuals are evaluated according to the fitness function. Then, using a selection strategy, the individuals with the highest fitness values are selected. Such a mechanism allows the best solutions to be retained while introducing new variants through mutation, enabling the system to continuously adapt to changing conditions [63,67].

While evolutionary programming was originally applied to the Finite-State Machines, it was later expanded to incorporate other methods. In some studies, solutions are represented as side-effect machines (SEMs), and evolutionary programming allows them to be mutated by adding or removing states and modifying transitions [65]. Evolutionary programming allows for the exploration and creation of different configurations of automata, as well as their flexible adaptation to changing conditions or the requirements of a specific task. This makes it a versatile tool to support the development of intelligent systems and allows both the structure and behavior of models to be automatically shaped in response to new challenges. Evolutionary programming can be used to generate new fuzzy rules or refine existing ones, ensuring that the system adapts to changing data distributions or operational requirements [63]. The authors of a related study verified the method using the example of iris set classification, demonstrating its effectiveness and applicability in a range of classification and diagnostic tasks [63].

2.5.3. Learning Algorithms

Learning algorithms are computational techniques that enable machines to identify patterns, make predictions, or perform tasks based on data. These algorithms form the basis of ML and AI, with applications in areas such as image recognition, robotics, and autonomous systems [68]. These algorithms can be supervised or unsupervised [64,65,66] and aim to update fuzzy membership functions, rule weights, or rule structures based on observed patterns or desired system behaviors [69].

Supervised learning algorithms, involving methods such as decision trees, neural networks, or support vector machines (SVMs), as well as linear and logistic regressions, can be used to modify fuzzy sets. By learning from input data with assigned labels, these algorithms are able to adjust membership functions or other parameters of fuzzy sets to best represent the patterns contained in the learning data [70,71,72]. For example, in hybrid neuro-fuzzy systems, neural networks automatically learn both the shapes of membership functions and optimize fuzzy rules, thereby combining the interpretability of fuzzy systems with the high adaptability of neural models. Decision trees can in turn be used to automatically generate transparent fuzzy rules based on the tree structure, while SVMs can support feature selection or parameter tuning by maximizing the margin of separation between classes. This approach enables models that are not only efficient but also robust to uncertainty and easy to interpret, which is particularly important in applications requiring decision explainability, such as medicine or finance.

Unsupervised learning algorithms, also referred to as learning without a teacher, find application in fuzzy set adaptation. These methods, such as K-Means, operate on unlabeled data, identifying hidden structures in the dataset, allowing the membership function or other parameters of fuzzy sets to be adapted in a way that reflects the internal organization of the input data [70,71,72]. This approach is particularly useful in situations where there is a lack of access to data with assigned class labels or where the aim is to discover new, non-obvious relationships in the set under analysis. The integration of unsupervised learning with fuzzy systems allows for the construction of models that are more flexible and adaptable in dynamic, uncertain environments.

In the field of learning algorithms, different classes of algorithms are distinguished, depending on the way they are taught and the type of problems they solve. In the following section, a selection of algorithms is presented within the three groups, along with their corresponding mathematical formulas:

(a)

Regression and classification algorithms (basic)—Classical approaches to regression and classification problems fall into this group. They are commonly used for linear or slightly non-linear data (

Table 1. Key regression and classification algorithms and their basic formulas [73].

Algorithm	Core Formula	Purpose
Linear Regression	$y={\beta }_0+{\beta }_{1}x+\epsilon$	Predict continuous values
Logistic Regression	$E\left(y\right)={\displaystyle \frac{1}{1+e^{-x\prime \beta }}}$	Binary classification

) [73].

where:

• y—predicted value (dependent variable);
• x—independent variable (feature);
• β₀, β₁—parameters of the linear regression model;
• $\epsilon$—random error component;
• E(y)—expected value of y, this is most commonly interpreted as the predicted probability of an event occurring y = 1;
• x′β—represents the scalar product of the transpose vector x and the parameter vector β;
• x^T—transposed feature vector.

(b)

Classical classification and clustering algorithms—This category includes algorithms used for both classification and unsupervised clustering of data (

Table 2. Classical classification and clustering algorithms and their basic formulas [74,75,76].

Algorithm	Core Formula	Purpose
SVM	Primal: $\underset{\beta , {\beta }_0}{\min }{\displaystyle \frac{1}{2}}{\left|\left|\beta \right|\right|}^2+C{\displaystyle \sum _{i=1}^N}{\xi }_i$ Subject to: ${\xi }_i\ge 0, y_i\left(x_i^T\beta +{\beta }_0\right)\ge 1-{\xi }_i$	Finding optional decision boundaries
Decision Trees	$Gini Impurity=1-{\displaystyle \sum _{i=1}^c}{P}_i^2$	Creating decision hierarchies
K-Means	$J={\displaystyle \sum _{n=1}^N}{\displaystyle \sum _{k=1}^K}{r}_{nk}\left|\right|x_n{-{\mu }_k\left|\right|}^2$	Discovering data clusters

) [74,75,76].

where:

• β—weight (parameter) vector;
• $\xi$—slack variable (SVM);
• C—regularization parameter (SVM);
• N—number of samples;
• y_i—class label for sample ii;
• p_i—proportion of class i in a node (Decision Trees);
• c—number of classes;
• J—objective function (K-Means);
• r_nk—assignment of point n o cluster k;
• x_n—data point;
• ${\mu }_k$—centroid of cluster k;
• K—number of clusters.

(c)

Advanced algorithm—The last group includes algorithms based on neural networks, which are more complex and used in dynamic environments or when processing large data sets. An example of such an algorithm is the neural network, whose core formula is a = f(W_p + b), where:

• neuron activation;
• f—activation function;
• W_p—weight of the connection in a neural network;
• bias;

allowing it to learn complex representations [77,78].

2.5.4. Reinforcement Learning

Reinforcement learning (RL), in combination with fuzzy logic-based systems, is an increasingly popular approach in the design of adaptive and explainable AI systems [78,79]. Through this combination, RL is able to optimize decision-making through trial and error, while fuzzy logic enables uncertainty modeling and knowledge representation through human-interpretable rules [80,81,82]. In this view, RL methods, such as the State–Action–Reward–State–Action (SARSA) algorithm, can be used to dynamically adjust the parameters of the membership function and the rules of a fuzzy system [83]. This process involves exploring different actions in a changing environment and modifying the system’s behavior based on the feedback received in the form of rewards. This approach allows for the development of intelligent systems that can not only adapt to changing and uncertain conditions but also retain the ability to generate transparent and interpretable decisions. In addition, the integration of RL with fuzzy logic makes it possible to create systems that are robust to noise and imprecise data, while allowing for easier modification of expert knowledge through fuzzy rule editing [79]. As a result, such hybrid solutions find applications in advanced control systems, autonomous robots, intelligent energy management systems, or adaptive decision support systems, where both adaptability and interpretability of decisions are crucial.

In RL, there are the following three major components [84]:

(a)

The environment—This is the task or simulation with which the algorithm (also called the agent or player) interacts. The goal of RL is to maximize the reward provided by the environment, i.e., to train the agent to achieve the maximum outcome in the environment, e.g., winning the most games or achieving the highest reward.

(b)

Agent—This is the element that interacts with the environment. The goal of the agent is to maximize the reward, i.e., to learn the most beneficial interaction with the environment. The agent’s behavior is determined by a so-called policy or function that is designed to return an appropriate action. Most commonly, a neural network is used as a policy.

(c)

Buffer—This is a data store that holds information collected by the agent during learning, which is then used to train the agent.

RL relies on a mathematical framework to optimize decision-making, with Bellman’s equation and the Q-learning update rule serving as its basic formulas. These equations allow agents to balance immediate rewards with long-term goals through iterative learning processes. The Bellman optimality equation can be represented by the following formula [85]:

$$v_{*}\left(s\right)=\underset{a\in A\left(s\right)}{\max }{\sum }_{s^{\prime }, r}p\left(s^{\prime }, r|s,a\right)[r+\gamma v_{*}\left(s^{\prime }\right)],$$

where:

• v_*(s) is the maximum expected value that can be obtained starting from state s and making optimal decisions;
• s—is the current state;
• a—the action taken in state s;
• A(s))—the set of all possible actions in state s;
• s′—the next state after performing the action a (next state);
• r—the reward received after moving to state s′ (reward);
• $p\left(s^{\prime },r|s,a\right)$—probability that after performing action a in state s in order to move to state s′ and receive reward r (transition probability);
• $\gamma$—discount factor, a number between 0 and 1, which determines how much future rewards count (discount factor) [83].

Q-learning, as a key RL algorithm, uses the Q-value update rule as follows [77]:

$$\mathrm{Q}\left(\mathrm{s},\mathrm{a}\right)\mathrm{Q}\left(\mathrm{s},\mathrm{a}\right)+\mathsf{\alpha }[\mathrm{R}+\mathsf{\gamma }\underset{a^{\prime }}{\max } Q\left({\mathrm{s}}^{\prime },{\mathrm{a}}^{\prime }\right)-\mathrm{Q}(\mathrm{s},\mathrm{a}\left)\right],$$

where:

• Q (s, a)—Q-value for state s and action a;
• α—learning rate;
• R—reward;
• $\mathsf{\gamma }$—discount factor;
• s, s′—current and next state;
• a, a′—current and next action.

2.5.5. Online Adaptation

Online adaptation is crucial for real-time systems. It allows the parameters of the system to be adjusted dynamically without the need for re-training. This is particularly important in applications such as financial market analysis or autonomous control [86,87,88]. In online adaptation methods, the fuzzy rules and parameters of the information system are continuously updated during operation. Such approaches typically use online learning algorithms or adaptive mechanisms that incorporate new information or change the system’s behavior in real time in response to changing inputs and outputs [89,90]. Online adaptation in fuzzy sets is particularly applicable to systems operating in dynamic and changing environments, where it is necessary to react quickly to new situations. Examples include share price prediction systems, the adaptive control of autonomous vehicles, or intelligent anomaly detection systems for critical infrastructure. For this purpose, special adaptation algorithms dedicated to fuzzy sets have also been developed and can be applied online [91,92], minimizing the risk of model obsolescence and ensuring the continuity and reliability of system operation. These solutions are often characterized by high computational efficiency and the ability to operate in resource-constrained environments, further increasing their practical usefulness.

The most common mathematical foundation for online adaptation is the incremental parameter update. The key formula takes the following form [76]:

$$w^{(\tau +1)}=w^{\tau }+\eta (t_n-w^{\left(\tau \right)T}{\varphi }_n){\varphi }_n$$

where $\tau$ denotes the iteration number; $\eta$ is a learning rate parameter; t_n is the target value; and ${\varphi }_n$ is the vector of features (basis functions) for the n-th sample. The value of w is initialized to some starting vector w⁽⁰⁾. This algorithm is called the least-mean-squares algorithm or the LMS algorithm [76].

2.5.6. Critical Comparison of Adaptation Approaches

GAs are characterized by high flexibility and high efficiency in solving optimization problems, especially in cases where the solution space is extensive and complex. Through the use of mechanisms such as crossover and mutation, they enable efficient searching of the solution space and increase the probability of finding the global optimum. However, it should be noted that their efficiency often involves the need to perform many iterations and maintain large populations of solutions, which generates significant demands on computational resources and increases the running time, especially for more complex tasks. Evolutionary programming, a development of the genetic approach, is proving more efficient in many applications, especially where dynamic adaptation of the solution structure or simultaneous optimization of multiple objectives is required. This method, although characterized by increased versatility and adaptability, also tends to be resource-intensive in terms of computation and may require the advanced implementation and design of appropriate matching functions.

Learning algorithms, such as neural networks, exhibit high performance in tasks where extensive learning datasets are available. Their main advantage is their ability to automatically detect complex relationships and adapt to new patterns, resulting in high prediction accuracy. On the other hand, they require significant time investment in the training process, as well as appropriately selected parameters to avoid overlearning. An additional challenge is the often limited interpretability of the resulting models.

RL is an effective tool in the context of tasks requiring sequential decision-making and adaptation to dynamically changing environmental conditions. This method allows for the gradual improvement of action strategies by maximizing long-term reward. However, its effectiveness is strongly dependent on the precise definition of the reward function and the appropriate balance of exploration and exploitation, which in practice may require a significant number of trials and prolong the learning process.

Online adaptation approaches, on the other hand, are particularly valuable in applications that require immediate response to changes in the environment and ongoing updating of model parameters. They enable real-time adaptation of the system, which increases its flexibility and usability in dynamic environments. However, it is important to bear in mind that rapid adaptation can lead to unstable performance, especially under conditions of rapid change or the presence of significant noise in the input data.

2.6. Tools and Resources for OFN Systems

The standardization of tools and benchmarks for systems based on OFNs is still in the early stages, but practical solutions already exist to support the implementation and testing of such models. An example is the proprietary libraries in the C# language (currently at alpha version level), which have been used to perform a number of calculations and experiments across various studies, including the work of Prokopowicz et al. [93]. Thanks to the simple structure of the OFN model, many calculations and analyses can also be performed and verified in popular spreadsheets such as Microsoft Excel. It is also worth noting that popular computing environments, such as Matlab (Fuzzy Logic Toolbox), jFuzzyLogic (Java) or scikit-fuzzy (Python), can be successfully extended with OFN-related functionalities, enabling experiments and implementations in different software environments [94]. The presence of these tools and application examples provides important support for the practical implementations of OFN systems and can be helpful for researchers and practitioners looking for ready-made solutions or inspiration for their own implementations. At the same time, the need for further development and standardization of tools, frameworks, and datasets dedicated to this class of systems should be emphasized as an important direction for future research.

2.7. Evolution of the OFN Model—Overview of Development Stages

After discussing the role of OFNs, challenges, development prospects, as well as presenting various adaptation methods, it is natural to examine how the OFN model has evolved over the years. The development of the OFN model has proceeded in stages, from early theoretical concepts to widespread applications in modern computational methods. The key phases of this evolution are outlined below (

Table 3. Evolution of OFN model.

Years	Key Developments
2002–2006	Conceptual phase and first implementations - Formal definition of OFN and first algebraic operations [1]; - Emergence of the idea to use OFNs in fuzzy systems [1]; - First OFN computing software implementation presented at the conference [17].
2007–2014	Development of methods and new applications - Intensive work on aggregation and adaptation methods for OFNs [95]; - First applications in economics and medical data evaluation [16,21]; - Hybridization with other computational methods [6].
2015–2020	Advanced trend processing and algorithm refinement - Development of trend-sensitive algorithms [4]; - Construction of complete fuzzy systems incorporating OFNs [6]; - Application to financial time series and robust modeling [96].
2021–present	New directions and broad applications - Use of OFNs in swarm intelligence (e.g., ant algorithms, pig herd algorithms) [37]; - Adaptation mechanisms for OFNs in neural networks and fuzzy systems [35]; - Application to high-frequency financial data and machine learning [3].

).

3. Discussion

This paper presents promising opportunities for enhancing the performance and capabilities of fuzzy logic applications across various domains. This review has highlighted several key aspects and potential directions for future development. OFNs allow for more flexible representation of uncertainty compared to traditional fuzzy numbers [14,18,19,20]. The directionality characteristic of OFNs enables the modeling of trends and tendencies, which is particularly valuable in dynamic systems. Such activities can lead to more accurate and complex phenomena, potentially improving the performance of fuzzy systems in areas such as financial forecasting, control systems, and decision support [21,30,31]. Well-defined OFN arithmetic operations that preserve directionality [11] offer advantages in terms of computational efficiency and interpretability [10,25,28]. This can lead to simplified rule bases and reduced complexity in fuzzy systems [10]. The potential for more efficient computing may make OFN-based fuzzy systems more suitable for real-time applications and large-scale data processing [35]. Despite its many advantages, several challenges remain. One major concern is the computational complexity associated with OFN processing, especially in large-scale applications [11]. Optimization techniques and efficient algorithms require further development to ensure that the benefits of OFNs do not come at the expense of impractical computation times. In addition, the standardization of methodologies for constructing and using OFNs remains an ongoing challenge, as various studies propose different approaches without a widely accepted framework. Despite numerous publications on OFNs, no universally accepted theoretical and practical framework for the concept has yet been created. This is probably because the history of OFN research is relatively short, and the topic that combines elements of mathematics, computer science, and decision theory. Looking into the future, the integration of OFN with AI offers promising opportunities for the development of adaptive fuzzy systems. Hybrid models that use deep learning techniques along with OFN-based reasoning could lead to smarter and more context-aware decision-making systems [3]. Moreover, interdisciplinary research combining OFNs with areas such as neuro-fuzzy systems and quantum computing may yield new insights into improving adaptive capabilities and predictive accuracy [8,52,97,98].

In order to deepen the analysis of adaptation methods in the context of the previously presented OFN concept, a comparison of the most relevant adaptation approaches was performed.

Table 4. Comparison of most relevant adaptations [99,100,101].

Method	Feature	Application	Advantages	Limitations
Genetic algorithms	Automatic adaptation of parameters (e.g., crossover and mutation rates) based on population information.	Parameter optimization, adaptation of fuzzy systems to achieve optimal solutions, parameterization of membership functions	Global search of the solution space, flexibility to adapt to change.	High computational complexity, slow convergence in large spaces.
Evolutionary programming	It is based on mutation as the main operator.	Search for optimal solutions, used to optimize parameters, improve system performance	Efficacy in continuous environments, no recombination required.	Slower convergence than GAs in some tasks.
Learning algorithms	Dynamic update of learning rate and model parameters	Classification, pattern recognition, data analysis	Effectiveness in environments with unstructured data.	Dependence on quality of input data, risk of overfitting.
Reinforcement learning	Reward/punishment-based learning from environment.	Adaptation of fuzzy system parameters and rules, optimization of fuzzy systems in dynamic and uncertain environments, decision making	Adaptation to unknown conditions, long-term optimization.	Requires big data resources, convergence problems in non-stationary environments.
Online adaptation	Fast adjustment of parameters without re-training	Monitoring systems, dynamic and changing environment	Computational efficiency, millisecond response.	Requires an initial training phase, instability in highly dynamic environments.

presents their features, typical applications, and main advantages and limitations, allowing for a better understanding of their practical potential. As shown in the table, GAs and evolutionary programming are particularly effective for global optimization, but they may suffer from high computational complexity and slow convergence. Learning algorithms and RL offer dynamic adaptation in complex and unstructured environments, though they require large amounts of data and may be prone to overfitting or convergence issues. Online adaptation stands out for its computational efficiency and real-time response, but it can be unstable in highly dynamic settings and still requires an initial training phase. This comparison illustrates that the choice of adaptation method should be carefully matched with the specific requirements and constraints of the application domain, balancing flexibility, computational cost, and robustness.

In addition to the summary of key adaptation methods, it is worth exploring the ways in which fuzzy systems based on OFNs can be adapted, as shown in

Table 5. Adaptation methods for fuzzy systems based on OFNs [3,8,30].

Method	Application Area
Variants of error propagation in neural networks with OFN	Learning neural networks, prediction
Adaptation of rules in control systems	Control of industrial processes
Adaptation of OFN discretization parameters	Calculation optimization, process modeling
Adjustment of cost thresholds in the LOB to OFN transformation	Analysis of stock market data, predictive systems
Adaptation of neural network architecture	Neuro-fuzzy network design
Gradient learning in OFN space	Machine learning, optimization
Context-dependent arithmetic (directional)	Simulation of dynamic systems, control

.

A comparison of different approaches to fuzzy sets (

Table 6. Comparison of OFNs and alternative fuzzy paradigms (own elaboration).

Aspect	OFNs	Type-1 Fuzzy Sets	Type-2 Fuzzy Sets	Intuitionistic Fuzzy Sets
Definition	Fuzzy numbers with explicit order and directionality	Standard fuzzy sets with a single membership function	Fuzzy sets where membership is another fuzzy set	Fuzzy sets with membership and non-membership function
Mathematical complexity	Moderate	Low	High	Moderate
Membership function	Describes both value and its orientation (increasing/decreasing)	Fixed membership function	Fluctuating Fuzzy membership	Membership + non-membership ≤ 1
Representative of uncertainty	Directional uncertainty (increasing/decreasing trends)	Limited to membership degree	Handles both value and uncertainty in membership function	Captures hesitation margin
Arithmetic operations	Well suited, flexible, preserves order and direction in computations	Simple but can lose details in complex operations	Complex but accurate in modeling uncertainty	Requires more complex arithmetic than Type-1 fuzzy sets
Interpretability	Medium, depending on the context of the data due to directional features	High: easy to understand and interpret	Low due to complex interpretation and implementation	Medium: requires understanding of hesitation degree
Advantages	Effectively models order, direction and uncertainty	Simple, easy to interpret and implement	Flexible, comprehensively captures uncertainty	Captures hesitation and dual uncertainty
Limitations	Less intuitive than Type-1 fuzzy sets	Cannot represent higher-order uncertainty	High computational demands	Can be hard to elicit both membership and non-membership data
Standardization	Still evolving in theory and applications	Standardized	Standardized but less implemented due to complexity	Less standardized—it varies across application domain
Adaptation	Emerging, increasingly used	Most widely used	Growing interest in high-uncertainty environments	Increasing use in certain applications (e.g., cognitive modeling)
Application(s)	Engineering, economics, decision making, trend modeling	Control systems, pattern recognition, classification	Control and decision making in highly uncertain environments	Medical diagnosis, human-related patterns recognition

)reveals the following:

• OFNs offer a nuanced extension of fuzzy numbers that preserves the order and direction of the trend, making them suitable for applications where directional uncertainty matters (e.g., dynamic systems, economics).
• Type-1 fuzzy sets are the basis of fuzzy logic, valued for their simplicity and efficiency, but limited in handling complex or vague uncertainty.
• Type-2 fuzzy sets introduce a higher order of flexibility and are efficient in uncertain environments, but their computational cost limits practical applications.
• Intuitionistic fuzzy sets are excellent at modeling fluctuations and dual perspectives (membership vs. non-membership)and are well suited for decision-making and human-centered systems.

3.1. Limitations of the Study

The literature on strictly OFN-based system adaptations is relatively narrow but provides a solid theoretical and methodological basis in the area of fuzzy systems, OFN arithmetic, and adaptation methods used in fuzzy systems research.

It is worth emphasizing that each of the methods discussed is applicable to different types of problems, so their selection should depend on the nature and type of data available. Despite numerous studies on the adaptation of knowledge systems, there are still noticeable gaps in the area related to fuzzy systems. Due to the wide possibilities of their use in various fields, it seems particularly important to develop this topic. Fuzzy systems play an important role especially in dynamic situations where data can change. Moreover, they have the advantage of being able to present information in a simple way, which facilitates communication even with non-specialists.

However, it should be noted that the generalizability of existing adaptation techniques based on fuzzy systems, including those using OFNs, remains a significant challenge. Many methods are designed and validated within specific application domains, and their direct transfer to other fields—e.g., from finance to medicine or engineering—may be limited by differences in data characteristics, domain requirements, or the availability of expert knowledge. Furthermore, the performance and robustness of OFN-based adaptation methods are highly sensitive to the choice of fuzzification strategies and parameters. Different approaches to input fuzzification can lead to substantial variations in outcomes, affecting both the accuracy and interpretability of the resulting models. The lack of standardization in this area makes it difficult to objectively compare results and ensure reliable transferability of methods across domains.

It is important to emphasize that the reviewed literature still reveals significant inconsistencies regarding both the definition of basic concepts and operations on OFN, as well as the criteria for evaluating the performance and interpretability of developed methods. Differences in formalization, such as divergent approaches to OFN parameters, lead to difficulties in direct comparison of results and limit the possibility of establishing universal standards. Furthermore, the lack of widely accepted measures of interpretability and performance means that the assessment of the effectiveness of individual solutions is often subjective and depends on the adopted assumptions.

Therefore, one of the key challenges for the further development of this field remains the standardization of definitions, formal criteria, and evaluation tools for OFN-based methods, which would enable more objective comparisons and contribute to greater transparency and practical utility of these solutions.

3.2. Key Directions for Future Research

Future research should focus not only on overcoming limitations but also on improving methodologies and integrating OFNs with new technologies. In particular, it is worth considering hybrid approaches that combine OFNs with new methods, which can lead to more advanced and efficient smart systems. With these challenges in mind, OFNs should be used with the goalof increasing the adaptability and robustness of fuzzy systems, making them more efficient and widely applicable. Along with advances in mathematics and computer science, an interdisciplinary approach will be a very important element in unlocking the potential of OFNs, ensuring a long-term impact on the evolution of intelligent systems.

An analysis of the current literature indicates that research directions such as the development of methods for the automatic calculation of OFN parameters, which will reduce the influence of subjective expert decisions and increase the repeatability of results, require particular attention. It is also important to apply OFNs in real-time systems and in environments with high data variability and uncertainty, such as autonomous control, online diagnostics or intelligent decision support systems. The literature also emphasizes the need for extensive comparative studies of different OFN adaptation methods on real-world datasets to identify their strengths and weaknesses in practice and to determine best practices in specific applications. Another important direction is the integration of OFNs with advanced machine learning techniques, including deep neural networks and evolutionary algorithms, in order to achieve more flexible and resilient systems. The creation of universal formal standards and software tools to facilitate the deployment of OFNs in different scientific and industrial domains also seems crucial.

Future research should focus on the practical challenges identified in the literature review, and the proposed developments aim to both advance theoretical knowledge and increase the application potential of OFNs.

4. Conclusions

This review has presented a comprehensive overview of the recent advances in adaptive techniques for systems based on OFNs. Over the past few years, one can observe the rapid development of adaptive methodologies that, when combined with OFNs, have enabled significant improvements in various engineering or scientific fields. The integration of OFNs with adaptive methods has led to significant improvements in handling uncertainty and nonlinearity across various domains, including control systems, time series forecasting, pattern recognition, and decision support in an environment of incomplete information. This review has identified both the advantages and challenges of OFNs, highlighting their important applications in both complex and dynamic environments. Foremost among the major benefits is the ability of OFNs to better represent imprecise information. However, despite the numerous advantages, there are also challenges, such as the need to develop more efficient computational methods or work on integration with fuzzy control systems. Consequently, the development of analytical tools and algorithms to optimize OFN performance is a key direction for future research.

The existing studies, however, do not lend themselves to full exploitation; in many cases, they require considerable development, modification, or detailing in the context of OFNs. The extent of these developments may be significant enough to warrant a separate group of methods dedicated to this type of representation.

Collaboration between experts from different fields—such as mathematics, computer science, engineering, and social sciences—can lead to the development of innovative solutions that can be applied to a wide range of practical problems. In the long term, the development of OFNs will contribute to increasing the efficiency and precision of intelligent systems, which will influence the advancement of adaptive technologies and their implementation in real-world applications.

It certainly seems important to develop new, more efficient adaptive algorithms dedicated to OFNs to better exploit their unique properties in practical applications. Particular attention should be given to research on the automation of the parameter selection and calculation process, which can significantly reduce subjectivity and increase the repeatability of results. In addition, it is worth conducting extensive comparative studies to assess the effectiveness of different OFN adaptation approaches on real datasets and in different application contexts.

Another important area is the integration of OFNs with modern machine learning techniques, including deep neural networks and evolutionary algorithms, which can contribute to more flexible and resilient fuzzy systems. It is also worth pursuing the development of universal formal standards and software tools that will facilitate the implementation of OFNs in various scientific and industrial fields. Advancing these research directions will not only contribute to theoretical knowledge, but, more importantly, increase the application potential of OFNs in next-generation intelligent systems.