Hierarchical Type-1 Mamdani Fuzzy System

• Hierarchical Type-1 Mamdani fuzzy systems are rule-based architectures that structure multiple fuzzy inference layers to manage complex, high-dimensional problems.
• They decompose the global input space into smaller subsystems, mitigating the curse of dimensionality and enhancing interpretability.
• The modular design enables practical control and prediction applications through advanced optimization techniques and integration with learning frameworks.

A Hierarchical Type-1 Mamdani Fuzzy System is a rule-based computational intelligence architecture that organizes multiple Mamdani fuzzy inference systems into a layered or tree-structured hierarchy. Such systems address the complexity, scalability, and interpretability challenges encountered in high-dimensional fuzzy modeling and control. Each fuzzy subsystem operates with conventional Type-1 Mamdani principles—membership functions, fuzzy rulebases, and defuzzification—but the interconnection and organization of these subsystems introduce additional design, optimization, and theoretical considerations.

1. Foundational Concepts and Motivations

A Type-1 Mamdani fuzzy system implements rules of the form “IF $X_1$ is $A_1$ AND $X_2$ is $A_2$ THEN $Y$ is $B$”, where $A_i$ and $B$ are fuzzy sets characterized by Type-1 membership functions $\mu_A(x): X \to [0,1]$. These systems rely on fuzzification, a rule base, fuzzy inference (typically using min/max or other t-norms for aggregation), and defuzzification (such as the centroid method) to map crisp inputs into crisp outputs.

As the number of input variables ($P$) and input fuzzy sets per variable ($k$) increases, the rule base size grows exponentially ($k^P$), leading to the "curse of dimensionality" and challenges in computational cost, maintenance, and interpretability (Ojha et al., 2019). Hierarchical Type-1 Mamdani fuzzy systems mitigate these issues by decomposing the global input space into smaller, coupled subsystems, each handling a subset of the inputs and aggregating their results through intermediate variables (Ojha et al., 2017, Ojha et al., 2019).

2. Hierarchical System Architecture and Design

Hierarchical fuzzy systems (HFS) are constructed by stacking multiple low-dimensional fuzzy logic units (FLUs), each realized as a Mamdani FIS, into a layered or tree-like structure (Ojha et al., 2019). The outputs of lower-level FLUs feed into higher-level units, creating a multi-stage mapping from original inputs to final outputs.

Two archetypal organizational patterns are prominent:

• Cascaded or Incremental HFS: FLUs are chained so each stage processes a subset of variables and passes an aggregated result upwards.
• Tree-like HFS: Nodes (FLUs) at each level combine distinct subsets of inputs and the resulting intermediate variables are aggregated at higher tree nodes (as in hierarchical fuzzy inference trees, or HFITs) (Ojha et al., 2017).

The system’s overall function $f$ is then composed as $f = f^{(L)}(f^{(L-1)}(...f^{(1)}(\mathbf{x})...))$, where $f^{(l)}$ is the mapping at hierarchy level $l$, and $L$ the total depth.

3. Inference Mechanics and Computational Workflow

Each individual Mamdani FLU conducts fuzzification, rule evaluation, and defuzzification independently, usually as follows (Elamvazuthi et al., 2010, Agustian et al., 2022):

1. Fuzzification: Crisp inputs are mapped to degrees of membership using defined functions, such as triangular or trapezoidal forms:

$$\mu(x;a,b,c)= \begin{cases} 0 & x \leq a\ \frac{x-a}{b-a} & a<x\leq b\ \frac{c-x}{c-b} & b<x\leq c\ 0 & x\geq c \end{cases}$$

1. Rule Evaluation: Each rule’s firing strength is computed, typically by minimum or product aggregation of antecedent memberships.
2. Implication and Aggregation: Consequent fuzzy sets are clipped (by their firing strengths) and aggregated across rules using the maximum operator.
3. Defuzzification: The centroid method is commonly applied:

$$y^* = \frac{ \int y \, \mu(y) dy }{ \int \mu(y) dy }$$

In hierarchical systems, the output from one Mamdani FIS (either a crisp or fuzzified intermediate variable) is fed as input to the next layer. For modular or tree-like arrangements, this means that different branches of the hierarchy may process disjoint input subsets, converging in higher-level aggregations (Ojha et al., 2017, Agustian et al., 2022).

4. Optimization and Learning Approaches

The effective deployment of hierarchical Mamdani systems hinges on the optimal allocation of variables to FLUs, the choice of hierarchy structure, and the calibration of membership function parameters. Several optimization approaches have been documented:

• Multiobjective Genetic Programming (MOGP): Used in HFITs to balance the trade-off between model accuracy (e.g., minimized RMSE) and complexity (minimal free parameters) (Ojha et al., 2017). MOGP evolves both the structure (which inputs to aggregate and how) and the rule base at each node.
• Differential Evolution (DE): Applied to tune membership function parameters at each fuzzy node. DE iteratively improves candidate solutions using population-based stochastic search, suitable even for non-differentiable objective functions (Ojha et al., 2017).
• Adaptive Neuro-Fuzzy Learning: In neuro-fuzzy controllers (e.g., for biped robots), parameter adjustment is conducted through gradient or hybrid optimization using training data (Zaidi et al., 2014).

Recent developments have leveraged deep learning frameworks to embed hierarchical FISs within end-to-end optimization architectures. This includes unconstrained parameterizations (e.g., using $\operatorname{sig}$ for membership heights) and GPU-accelerated, mini-batched inference to address computational bottlenecks (Koklu et al., 19 Apr 2024).

5. Applications Across Domains

Hierarchical Type-1 Mamdani fuzzy systems have demonstrated efficacy across a spectrum of real-world problems:

• Robotic Control: Hierarchical neuro-fuzzy systems have been used to control biped walking by modularizing joint control based around the robot’s center of mass, with individual Mamdani-like sub-controllers maintaining the stability and coordination of each leg (Zaidi et al., 2014).
• Industrial Control and Agriculture: In NFT hydroponic systems, a hierarchical Mamdani fuzzy controller normalizes pH and TDS through low-level closed-loop control (local sensors and actuators) and high-level IoT-based oversight (Agustian et al., 2022).
• Function Approximation and Time Series Prediction: HFITs efficiently decompose high-dimensional system identification tasks, yielding models with low RMSE and reduced rule base size compared to traditional “flat” fuzzy systems (Ojha et al., 2017).
• Social Science and Survey Analysis: A hierarchical Mamdani system aggregates Likert-scale evaluations of project success across multiple dimensions, enabling a nuanced and context-sensitive measure of overall project performance (Granja-Correia et al., 16 Jul 2025).

6. Theoretical Extensions and Efficiency Enhancements

The hierarchical architecture is not limited to standard Mamdani operations. Several innovations and theoretical advances are notable:

• Law of Importation (LIA) and Modular Inference: Certain aggregation functions and fuzzy implications satisfy the law of importation,

$I(A(x, y), z) = I(x, I(y, z))$

enabling sequential decomposition of multi-input inference into hierarchical steps and drastically reducing computational complexity (Li et al., 2021).

• Logical Embedding and Automated Reasoning: Mamdani rule systems have been formalized in Gödel logic with truth constants, supporting the deduction of system properties (e.g., reachability, stability) through clausal form translation and hyperresolution calculus. This allows the analysis and verification of hierarchical fuzzy systems using automated theorem proving (Guller, 2023).
• Computational Learning via Deep Architectures: Hierarchical Mamdani systems can be implemented in deep learning frameworks where parameter learning and inference are vectorized and parallelized, allowing scalable training and real-time deployment even with deep hierarchies (Koklu et al., 19 Apr 2024).

7. Limitations, Challenges, and Prospects

Hierarchical Type-1 Mamdani systems offer significant reduction in rule base explosion, improved modularity, and scalability, but they also present several challenges:

• Inter-layer Coupling and Uncertainty Propagation: Information loss or uncertainty accumulation can occur as outputs propagate through layers, requiring careful design and possibly hybrid approaches with interval Type-2 or enhanced reasoning schemes.
• Interpretability of Intermediate Variables: With increased depth, the meaning of intermediate variables may become opaque, complicating expert validation and maintenance (Ojha et al., 2019).
• Structure Selection and Optimization: Automatic determination of optimal hierarchies, as well as input partitioning for each FLU, remains a complex problem, though recent evolutionary and learning-based methods show promise (Ojha et al., 2017, Koklu et al., 19 Apr 2024).
• Computational Costs in Deep Hierarchies: While hierarchical modularization alleviates exponential rule growth, it may introduce its own computational and memory costs when systems become very deep unless efficient parameterization and parallelization are adopted (Koklu et al., 19 Apr 2024).

A plausible implication is that continued integration of heuristic optimization, deep learning, modular hardware designs, and formal logical analysis will further enable the deployment of hierarchical Type-1 Mamdani fuzzy systems in large-scale, safety-critical, and context-sensitive applications across engineering and the social sciences.