Discrete Fuzzy Numbers: Theory & Applications

• Discrete fuzzy numbers are finite-domain fuzzy numbers defined by a closed support and monotonic membership functions.
• They enable robust decision-making through efficient ranking, aggregation, and arithmetic operations using total orderings and Zadeh’s extension principle.
• Their computational framework leverages bijections and interval-based techniques to achieve scalable ranking, defuzzification, and aggregation performance.

Discrete fuzzy numbers (dfns) formalize the concept of fuzzy numbers in the context of finite, discrete domains, enabling both rigorous arithmetic and scalable computational treatment for applications in fuzzy systems, decision-making, and linguistic modeling. Their definitions, ordering, and operations have been the subject of extensive paper, providing efficient frameworks for aggregation, implication, ranking, and defuzzification processes.

1. Mathematical Definition and Structure

A discrete fuzzy number is typically constructed as a mapping $A : L_n \to Y_m$ where:

• $L_n = \{0, 1, ..., n\}$ is a finite chain, serving as the carrier or support.
• $Y_m = \{0 = y_1 < y_2 < ... < y_m = 1\}$ is a finite set of admissible membership values.

The key properties defining a dfn $A \in \mathcal{D}_1^{L_n \rightarrow Y_m}$ are:

• Support as a closed subinterval: $\operatorname{supp}(A) = [s, t] \subseteq L_n$ for some $0 \leq s \leq t \leq n$.
• Core condition: $A(i) = 1$ for every $i \in [s, t]$.
• Nondecreasing on the left: For $0 \leq i < j \leq s$, $A(i) \leq A(j)$.
• Nonincreasing on the right: For $t \leq i < j \leq n$, $A(i) \geq A(j)$.

The total number of such discrete fuzzy numbers is $$|\mathcal{D}_1^{L_n \rightarrow Y_m}| = \binom{n+2m-2}{2m-2}$$, reflecting underlying combinatorial constraints (Mir et al., 21 Nov 2025).

2. Admissible Total Orderings

Comparing discrete fuzzy numbers requires establishing total orders on their supports. This typically leverages the set $I_n$ of all closed intervals in $L_n$, equipped with an admissible total order $\preceq$ that:

• Is total: Any pair $[a, b], [c, d]$ is comparable.
• Extends the partial (componentwise) order: If $a \leq c$ and $b \leq d$, then $[a, b] \preceq [c, d]$.

Canonical choices for $\preceq$ include lexicographic (by left or right endpoint), Xu–Yager, or "t-inc" (by left endpoint, then decreasing right endpoint).

A total order $\prec$ on the set of dfns is induced by comparing their $\gamma$-cuts at the level where two sequences first differ. If $A^\gamma$ and $B^\gamma$ differ at maximal $\gamma$, then $A \prec B \iff A^\gamma \preceq B^\gamma$. This order can be proven admissible and total (Mir et al., 21 Nov 2025).

3. Efficient Ranking and Unranking: The pos Function

A remarkable advancement is the construction of an efficient bijection—called the pos function—from the set of dfns to $\{0, ..., N-1\}$, where $N = |\mathcal{D}_1^{L_n \rightarrow Y_m}|$. Given the admissible total order, the pos function satisfies:

• $pos(A) = i$ iff $A$ is the $i$-th dfn in total order.
• The inverse $pos^{-1}(i)$ yields the unique $A$ of rank $i$.

By decomposing dfns via their $\alpha$-cuts and leveraging closed-form counts $|SDFN(a, b, j)|$ for the number of dfns with a fixed $\gamma$-cut, efficient $O(n^2 m \log n)$ algorithms for both ranking and unranking are achieved. This is a substantial reduction from the exponential time implied by explicit enumeration, allowing practical application for large $n$ or $m$ (Mir et al., 21 Nov 2025).

4. Extension Principles and Arithmetic

The arithmetic of discrete fuzzy numbers uses Zadeh’s extension principle, extended to discrete domains: $$\mu_{C}(z) = \bigvee_{x \circ y = z} \left( \mu_{A}(x) \wedge \mu_{B}(y) \right)$$ for any binary operator $\circ$, where $A$, $B$ are dfns, $C = A \circ B$ is the resulting dfn, and $\mu$ denotes their membership functions. This methodology extends to all arithmetic operations and to cardinal semantic operators, supporting not only crisp but also fuzzy parameters (e.g., fuzzy radix, fuzzy conversion rate) (Chunikhin et al., 2022).

Structured frameworks generalize the arithmetic further to membership and carrier lattices $(L, M)$, with the join (∨) and meet (∧) in $M$ replacing supremum and infimum in $[0, 1]$ for the evaluation of membership in $C$ (Maximov, 2022).

5. Aggregation, Defuzzification, and Weighted Averages

Defuzzification and aggregation are central in fuzzy decision-making. The Level-Based Weighted Average (WABL) is a flexible operator for discrete fuzzy numbers, particularly discrete trapezoidal and triangular variants: $$\mathrm{WABL}(A) = \sum_{i=0}^{t} P_i \left[ (1-c)L_A(\alpha_i) + cR_A(\alpha_i) \right]$$ where $P_i$ are level weights, $c\in[0,1]$ is an optimism parameter, and $L_A$, $R_A$ are the left/right endpoints of the $\alpha$-cuts at level $\alpha_i$. For standard weight patterns (constant, linear, quadratic), WABL reduces to simple closed forms involving only the endpoints at the lowest and highest levels, yielding exact, efficient computation for discrete fuzzy numbers (Nasiboglu et al., 2018).

6. Algorithmic and Computational Aspects

State-of-the-art algorithms for pos and $pos^{-1}$ exhibit $O(n^2 m \log n)$ complexity, with the dominant cost linear in the number of membership levels $m$. Empirical validation (e.g., for $n=10$, $m$ ranging from $100$ to $1000$) demonstrates linear scaling in $m$ and low per-operation times (on the order of milliseconds). Memory overhead is minimal beyond precomputing and storing the sorted interval list of size $O(n^2)$.

Algebraic operations (e.g., lattice aggregation, implication) on dfns can be canonically implemented by translating the operation to the corresponding operation on integer indices via the pos bijection, then mapping back. This stratagem sidesteps direct combinatorial explosion (Mir et al., 21 Nov 2025).

7. Applications and Generalizations

Discrete fuzzy numbers are integral in fuzzy-linguistic systems, cognitive maps, fuzzy numeration systems, and expert-based decision frameworks. In multi-valued cognitive maps, lattice-based dfns permit consensus estimation and robust aggregation of expert inputs, with convergence of updates governed by appropriate semilattice operations. Mean-value analogues (pessimistic/optimistic) have been formalized using join and meet operations on lattices (Maximov, 2022). In numeration systems, dfns underpin fully fuzzy generalizations of cardinal operations, enabling joint fuzziness in both digits and arithmetic operators (Chunikhin et al., 2022).

Applications leveraging the efficient pos framework can scale to high granularities of membership, enabling large vocabularies in fuzzy-linguistic models and precise, computationally tractable arithmetic in discrete environments.

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Key References:

• An Efficient Computational Framework for Discrete Fuzzy Numbers Based on Total Orders (Mir et al., 21 Nov 2025)
• Lattice Generalizations of the Concept of Fuzzy Numbers and Zadeh's Extension Principle (Maximov, 2022)
• On Fuzzy Cardinal Semantic Transformations (Chunikhin et al., 2022)
• Analytical Formulations for the Level Based Weighted Average Value of Discrete Trapezoidal Fuzzy Numbers (Nasiboglu et al., 2018)
• Arithmetic of fuzzy numbers and intervals -- a new perspective with examples (Schneider, 2013)