Adaptation of Fuzzy Systems Based on Ordered Fuzzy Numbers: A Review of Applications and Development Prospects
Abstract
:1. Introduction
2. Ordered Fuzzy Numbers
2.1. Definition of OFNs and Arithmetic Properties
- f (increasing part) represents the up-part of the fuzzy number;
- (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.
2.2. The Role of OFNs
2.3. Challanges
2.4. Development Prospects
2.4.1. Theoretical Developments
2.4.2. Practical Implementations
2.5. Adaptation Methods
2.5.1. Genetic Algorithms
2.5.2. Evolutionary Programming
2.5.3. Learning Algorithms
- (a)
- y—predicted value (dependent variable);
- x—independent variable (feature);
- β0, β1—parameters of the linear regression model;
- —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;
- xT—transposed feature vector.
- (b)
- β—weight (parameter) vector;
- —slack variable (SVM);
- C—regularization parameter (SVM);
- N—number of samples;
- yi—class label for sample ii;
- pi—proportion of class i in a node (Decision Trees);
- c—number of classes;
- J—objective function (K-Means);
- rnk—assignment of point n o cluster k;
- xn—data point;
- —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(Wp + b), where:
- neuron activation;
- f—activation function;
- Wp—weight of the connection in a neural network;
- bias;
2.5.4. Reinforcement Learning
- (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.
- 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);
- —probability that after performing action a in state s in order to move to state s′ and receive reward r (transition probability);
- —discount factor, a number between 0 and 1, which determines how much future rewards count (discount factor) [83].
- Q (s, a)—Q-value for state s and action a;
- α—learning rate;
- R—reward;
- —discount factor;
- s, s′—current and next state;
- a, a′—current and next action.
2.5.5. Online Adaptation
2.5.6. Critical Comparison of Adaptation Approaches
2.6. Tools and Resources for OFN Systems
2.7. Evolution of the OFN Model—Overview of Development Stages
3. Discussion
- 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
3.2. Key Directions for Future Research
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Algorithm | Core Formula | Purpose |
---|---|---|
Linear Regression | Predict continuous values | |
Logistic Regression | Binary classification |
Algorithm | Core Formula | Purpose |
---|---|---|
SVM | Primal: Subject to: | Finding optional decision boundaries |
Decision Trees | Creating decision hierarchies | |
K-Means | Discovering data clusters |
Years | Key Developments |
---|---|
2002–2006 | Conceptual phase and first implementations |
2007–2014 | Development of methods and new applications |
2015–2020 | Advanced trend processing and algorithm refinement |
2021–present | New directions and broad applications |
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. |
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 |
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 |
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Małolepsza, O.; Mikołajewski, D.; Prokopowicz, P. Adaptation of Fuzzy Systems Based on Ordered Fuzzy Numbers: A Review of Applications and Development Prospects. Electronics 2025, 14, 2341. https://doi.org/10.3390/electronics14122341
Małolepsza O, Mikołajewski D, Prokopowicz P. Adaptation of Fuzzy Systems Based on Ordered Fuzzy Numbers: A Review of Applications and Development Prospects. Electronics. 2025; 14(12):2341. https://doi.org/10.3390/electronics14122341
Chicago/Turabian StyleMałolepsza, Olga, Dariusz Mikołajewski, and Piotr Prokopowicz. 2025. "Adaptation of Fuzzy Systems Based on Ordered Fuzzy Numbers: A Review of Applications and Development Prospects" Electronics 14, no. 12: 2341. https://doi.org/10.3390/electronics14122341
APA StyleMałolepsza, O., Mikołajewski, D., & Prokopowicz, P. (2025). Adaptation of Fuzzy Systems Based on Ordered Fuzzy Numbers: A Review of Applications and Development Prospects. Electronics, 14(12), 2341. https://doi.org/10.3390/electronics14122341