Semi-Fuzzy Linear Systems

• Semi-Fuzzy Linear Systems are linear systems with a crisp coefficient matrix and fuzzy right-hand side or initial conditions, defining uncertainty via fuzzy numbers.
• They are analyzed using parametric-endpoint, geometric, and block-embedding methods that yield solution sets like parallelepipeds and convex polyhedra with clear existence and uniqueness criteria.
• Recent extensions incorporate fuzzy-number spaces and ring structures, broadening SFLS applications in both algebraic formulations and differential equation models.

Semi-Fuzzy Linear System (SFLS) denotes a linear system in which the coefficient matrix is crisp and the uncertainty is encoded by fuzzy numbers. In the algebraic setting, the standard model is $Ax=\tilde b$, where $A\in\mathbb R^{n\times n}$ is a crisp matrix and $\tilde b$ is a fuzzy $n$-vector; in the differential setting, the model is $x'(t)=Ax(t)+g(t)$, $x(0)=\tilde x_0$, with crisp $A$ and $g(t)$, and a fuzzy initial condition (Salkuyeh, 2014, 0910.4307). The literature does not impose a single solution concept. Some works seek a fuzzy $n$-vector in parametric form, while geometric approaches define the solution as a fuzzy set of crisp vectors, or a fuzzy set of real vector-functions, each member satisfying the underlying crisp system with a certain possibility (0910.4049, 0910.4307).

1. Definition, scope, and principal variants

In the usual SFLS setting for algebraic systems, the coefficient matrix is crisp and the right-hand side is fuzzy. Because $A$ is crisp and $A\in\mathbb R^{n\times n}$0 is fuzzy, this is a semi-fuzzy linear system; by contrast, a fully fuzzy linear system (FFLS) would have fuzzy coefficients in $A\in\mathbb R^{n\times n}$1 and a fuzzy right-hand side (0910.4049, Salkuyeh, 2014). The same crisp-versus-fuzzy separation appears in the differential-equation literature, where the dynamics are governed by a linear ordinary differential equation with crisp real coefficients and the fuzziness resides only in the initial condition (0910.4307).

The subject includes square, overdetermined, underdetermined, and rank-deficient systems. For $A\in\mathbb R^{n\times n}$2 with fuzzy right-hand side $A\in\mathbb R^{n\times n}$3, all possible cases pertaining to the number of variables $A\in\mathbb R^{n\times n}$4 and the number of equations $A\in\mathbb R^{n\times n}$5 have been treated: for $A\in\mathbb R^{n\times n}$6, the solution set is shown to be a parallelepiped in coordinate space; for $A\in\mathbb R^{n\times n}$7, the solution set is a convex polyhedron; and for $A\in\mathbb R^{n\times n}$8, the general solution is computed by determining the contribution of free variables (0911.0790).

A second distinction concerns the object called the “solution.” In the parametric-endpoint tradition, the unknown $A\in\mathbb R^{n\times n}$9 is a vector of fuzzy numbers, and the system is satisfied level-wise or endpoint-wise. In the geometric tradition, the solution is sought not as a vector of fuzzy numbers but as a fuzzy set of crisp vectors $\tilde b$0, with membership inherited from the fuzzy data through the crisp map $\tilde b$1 (0910.4049). For differential equations, the analogous shift is from a fuzzy vector-function to a fuzzy set of real vector-functions generated by crisp initial points inside the fuzzy initial region (0910.4307).

2. Fuzzy-number models and crisp embeddings

A recurrent representation is the parametric form $\tilde b$2, $\tilde b$3, where $\tilde b$4 is bounded, left-continuous, and nondecreasing, $\tilde b$5 is bounded, right-continuous, and nonincreasing, and $\tilde b$6 for all $\tilde b$7 (Salkuyeh, 2014). A crisp real $\tilde b$8 is the particular fuzzy number with $\tilde b$9. Triangular fuzzy numbers $n$0 and trapezoidal fuzzy numbers $n$1 are standard special cases. For a triangular fuzzy number, the $n$2-cut is

$n$3

and more general LR fuzzy numbers have componentwise $n$4-cuts of the form $n$5 (0910.4307). For a trapezoidal fuzzy number, the parametric endpoints are $n$6 and $n$7 (Salkuyeh, 2014).

To solve SFLS with a crisp coefficient matrix and a fuzzy unknown or fuzzy right-hand side, several papers use a $n$8 embedded crisp system obtained by separating positive and negative coefficients. One formulation defines $n$9 as the positive part of $x'(t)=Ax(t)+g(t)$0, $x'(t)=Ax(t)+g(t)$1, and

$x'(t)=Ax(t)+g(t)$2

where $x'(t)=Ax(t)+g(t)$3 and $x'(t)=Ax(t)+g(t)$4 stack endpoint vectors (Salkuyeh, 2014). A related formulation writes $x'(t)=Ax(t)+g(t)$5, with $x'(t)=Ax(t)+g(t)$6 and $x'(t)=Ax(t)+g(t)$7, and uses

$x'(t)=Ax(t)+g(t)$8

together with the coupled endpoint equations

$x'(t)=Ax(t)+g(t)$9

(Amrahov et al., 2011). In the core-EP inverse approach, the associated matrix has block form

$x(0)=\tilde x_0$0

where $x(0)=\tilde x_0$1 and $x(0)=\tilde x_0$2 collect the positive and negative parts of $x(0)=\tilde x_0$3, respectively (Gao et al., 2021).

These embeddings serve different purposes. In the matrix-class literature, they support existence and uniqueness theorems for fuzzy vector solutions (Salkuyeh, 2014). In the strong-solution literature, they isolate the endpoint-order condition $x(0)=\tilde x_0$4 (Amrahov et al., 2011). In the generalized-inverse literature, they reduce consistent and inconsistent fuzzy systems to crisp systems of doubled size (Gao et al., 2021).

3. Geometric solution sets for algebraic SFLS

A central geometric observation is that a vector of triangular fuzzy numbers forms an axis-aligned hyperrectangle in $x(0)=\tilde x_0$5, and the image of a parallelepiped is also a parallelepiped under a linear transformation (0910.4049). If $x(0)=\tilde x_0$6 has triangular components $x(0)=\tilde x_0$7, one writes $x(0)=\tilde x_0$8, $x(0)=\tilde x_0$9, $A$0, and

$A$1

For a nonsingular crisp matrix $A$2, the crisp “central” solution is $A$3, and the solution set is

$A$4

The $A$5-cut satisfies

$A$6

so every $A$7-cut is a parallelepiped centered at $A$8 (0910.4049).

In this framework the membership of a crisp vector is inherited from the right-hand side: $A$9 For triangular data, the construction can be parameterized by support vectors $g(t)$0 and $g(t)$1, writing

$g(t)$2

with membership

$g(t)$3

The same paper proves a necessary and sufficient characterization for when the solution can be represented as a vector of fuzzy numbers for any right-hand side: this happens if and only if $g(t)$4, where $g(t)$5 is a permutation matrix and $g(t)$6 is a nonsingular diagonal matrix, i.e. $g(t)$7 is a generalized permutation matrix (0910.4049).

The non-square theory extends the same geometric viewpoint. For $g(t)$8, the $g(t)$9-cut of the right-hand side is the box

$n$0

and the $n$1-cut of the solution set is

$n$2

For $n$3 and $n$4 invertible, $n$5 is a parallelepiped. For $n$6, each equation defines a slab and the intersection is a convex polyhedron. For $n$7, one partitions variables into leading and free variables, solves a square subsystem for the fuzzy part, and translates by the homogeneous contribution of the free variables (0911.0790). In all cases, the membership is

$n$8

which the paper identifies with the “united solution set (USS)” viewpoint (0911.0790).

A persistent misconception addressed by the geometric literature is that one may always force an SFLS solution into a componentwise vector of fuzzy numbers. The geometric proofs show that, unless $n$9 is a generalized permutation matrix, the image $A$0 is generally a rotated or sheared parallelepiped, so the solution is naturally a fuzzy set of crisp vectors rather than a coordinatewise fuzzy vector (0910.4049).

4. Strong solutions, existence, uniqueness, and matrix classes

When the unknown is required to be a vector of fuzzy numbers in parametric form, the distinction between strong and weak solutions becomes decisive. A strong fuzzy solution is a vector of fuzzy numbers $A$1 such that, for all $A$2, the endpoint functions satisfy $A$3, with $A$4 increasing and $A$5 decreasing; a weak solution is obtained when the embedded crisp system is solvable but, for at least one $A$6 and some $A$7, one has $A$8 (Amrahov et al., 2011).

For the embedding $A$9, $A\in\mathbb R^{n\times n}$00, the key separated variables are

$A\in\mathbb R^{n\times n}$01

which satisfy

$A\in\mathbb R^{n\times n}$02

Hence

$A\in\mathbb R^{n\times n}$03

Assuming $A\in\mathbb R^{n\times n}$04 is nonsingular, equivalently $A\in\mathbb R^{n\times n}$05 and $A\in\mathbb R^{n\times n}$06 are both nonsingular, the SFLS has a unique strong fuzzy solution if and only if

$A\in\mathbb R^{n\times n}$07

for all $A\in\mathbb R^{n\times n}$08 (Amrahov et al., 2011). The classical $A\in\mathbb R^{n\times n}$09-independent guarantee is $A\in\mathbb R^{n\times n}$10, equivalently $A\in\mathbb R^{n\times n}$11; the paper states that this forces $A\in\mathbb R^{n\times n}$12, where $A\in\mathbb R^{n\times n}$13 is a permutation matrix and $A\in\mathbb R^{n\times n}$14 is a nonsingular diagonal matrix, so the classical theorem applies only to a special form of linear systems (Amrahov et al., 2011).

A different existence–uniqueness line is based on matrix classes. For the Friedman–Ming–Kandel embedding

$A\in\mathbb R^{n\times n}$15

the block matrix $A\in\mathbb R^{n\times n}$16 is nonsingular if and only if both $A\in\mathbb R^{n\times n}$17 and $A\in\mathbb R^{n\times n}$18 are nonsingular. The unique solution $A\in\mathbb R^{n\times n}$19 yields valid fuzzy endpoints for arbitrary $A\in\mathbb R^{n\times n}$20 if and only if $A\in\mathbb R^{n\times n}$21 elementwise (Salkuyeh, 2014). Within this framework, if $A\in\mathbb R^{n\times n}$22 is an H-matrix, then there exists a permutation matrix $A\in\mathbb R^{n\times n}$23 such that $A\in\mathbb R^{n\times n}$24 is an H-matrix; consequently, $A\in\mathbb R^{n\times n}$25 is nonsingular and the embedded system has a unique solution for every $A\in\mathbb R^{n\times n}$26. If $A\in\mathbb R^{n\times n}$27 is an M-matrix, then $A\in\mathbb R^{n\times n}$28 is an M-matrix and $A\in\mathbb R^{n\times n}$29, so $A\in\mathbb R^{n\times n}$30 holds for arbitrary fuzzy $A\in\mathbb R^{n\times n}$31. Strictly diagonally dominant (SDD) matrices appear as a subclass of H-matrices, and the paper records corresponding permutation and diagonal-positivity corollaries for $A\in\mathbb R^{n\times n}$32 or $A\in\mathbb R^{n\times n}$33 (Salkuyeh, 2014).

For singular or inconsistent embedded systems, the core-EP inverse provides a generalized solution theory. With associated matrix $A\in\mathbb R^{n\times n}$34 and index $A\in\mathbb R^{n\times n}$35, Theorem 4.1 states that $A\in\mathbb R^{n\times n}$36 is a solution of $A\in\mathbb R^{n\times n}$37 if and only if $A\in\mathbb R^{n\times n}$38. Thus, if $A\in\mathbb R^{n\times n}$39, then $A\in\mathbb R^{n\times n}$40 is the unique solution; if $A\in\mathbb R^{n\times n}$41 and $A\in\mathbb R^{n\times n}$42, then $A\in\mathbb R^{n\times n}$43 is a solution (Gao et al., 2021). For the inconsistent case, the paper proposes generalized solutions through the consistent surrogate systems

$A\in\mathbb R^{n\times n}$44

for which the same $A\in\mathbb R^{n\times n}$45 is a solution. The paper does not explicitly attach a least-squares or minimum-norm optimality interpretation; rather, it proves that the same $A\in\mathbb R^{n\times n}$46 solves each of the consistent surrogate systems (Gao et al., 2021).

5. Semi-fuzzy linear systems of differential equations

For linear ordinary differential equations, the SFLS initial value problem is

$A\in\mathbb R^{n\times n}$47

where $A\in\mathbb R^{n\times n}$48 is crisp, $A\in\mathbb R^{n\times n}$49 is crisp, and $A\in\mathbb R^{n\times n}$50 is an $A\in\mathbb R^{n\times n}$51-vector of fuzzy numbers (0910.4307). The key conceptual shift is that the solution is not sought as a fuzzy vector-function $A\in\mathbb R^{n\times n}$52. Instead, it is a fuzzy set of real vector-functions, each trajectory satisfying the crisp ODE with some possibility. Each crisp initial point $A\in\mathbb R^{n\times n}$53 within the fuzzy initial region generates a unique crisp trajectory

$A\in\mathbb R^{n\times n}$54

and the possibility of that trajectory equals the membership of $A\in\mathbb R^{n\times n}$55 in the initial fuzzy set (0910.4307).

Let $A\in\mathbb R^{n\times n}$56. If the initial $A\in\mathbb R^{n\times n}$57-cut $A\in\mathbb R^{n\times n}$58 is written as a hyperrectangle, then in the homogeneous case $A\in\mathbb R^{n\times n}$59,

$A\in\mathbb R^{n\times n}$60

Since $A\in\mathbb R^{n\times n}$61 is linear and invertible for all $A\in\mathbb R^{n\times n}$62, the image of a hyperrectangle under $A\in\mathbb R^{n\times n}$63 is a parallelepiped, and as $A\in\mathbb R^{n\times n}$64 increases these are nested parallelepipeds. In the nonhomogeneous case,

$A\in\mathbb R^{n\times n}$65

so

$A\in\mathbb R^{n\times n}$66

At any time, the solution therefore constitutes a fuzzy region in the coordinate space, $A\in\mathbb R^{n\times n}$67-cuts of which are nested parallelepipeds (0910.4307).

The membership of a point $A\in\mathbb R^{n\times n}$68 at time $A\in\mathbb R^{n\times n}$69 is recovered from the $A\in\mathbb R^{n\times n}$70-cuts by

$A\in\mathbb R^{n\times n}$71

Operationally, one computes $A\in\mathbb R^{n\times n}$72 in the nonhomogeneous case, or $A\in\mathbb R^{n\times n}$73 in the homogeneous case, and tests componentwise inclusion of $A\in\mathbb R^{n\times n}$74 in the initial $A\in\mathbb R^{n\times n}$75-cut. For triangular initial numbers with modal values $A\in\mathbb R^{n\times n}$76, left spreads $A\in\mathbb R^{n\times n}$77, and right spreads $A\in\mathbb R^{n\times n}$78, the piecewise ratios

$A\in\mathbb R^{n\times n}$79

yield

$A\in\mathbb R^{n\times n}$80

The paper notes that using $A\in\mathbb R^{n\times n}$81 typically improves conditioning (0910.4307).

The worked two-dimensional “arms race” model illustrates the construction with

$A\in\mathbb R^{n\times n}$82

The matrix $A\in\mathbb R^{n\times n}$83 has eigenvalues $A\in\mathbb R^{n\times n}$84 and $A\in\mathbb R^{n\times n}$85, and the crisp center trajectory is

$A\in\mathbb R^{n\times n}$86

Thus the fuzzy solution at time $A\in\mathbb R^{n\times n}$87 is a parallelogram centered at $A\in\mathbb R^{n\times n}$88, spanned by the two columns of $A\in\mathbb R^{n\times n}$89, and scaled by $A\in\mathbb R^{n\times n}$90 along each generator. As $A\in\mathbb R^{n\times n}$91 increases, the parallelogram shrinks to a point at the origin as $A\in\mathbb R^{n\times n}$92 (0910.4307).

6. Recent extensions and specialized algebraic frameworks

Recent work has extended SFLS beyond triangular or parametric endpoint models by placing fuzzy numbers in spaces with explicit algebraic structure. In the Gaussian probability density membership function space $A\in\mathbb R^{n\times n}$93, a fuzzy number is parameterized by a 5-tuple $A\in\mathbb R^{n\times n}$94, and $A\in\mathbb R^{n\times n}$95 is both a 5-dimensional real vector space and a commutative ring with identity (Zheng, 21 Jul 2025). For the SFLS

$A\in\mathbb R^{n\times n}$96

the five coordinates $A\in\mathbb R^{n\times n}$97 each satisfy a classical real linear system with the same matrix $A\in\mathbb R^{n\times n}$98. The system is consistent if and only if $A\in\mathbb R^{n\times n}$99; if $\tilde b$00, the solution is unique; if $\tilde b$01, the solution set is a $\tilde b$02-dimensional affine space. For square $\tilde b$03, the paper presents Cramer’s rule in $\tilde b$04, and for RREF matrices it gives an explicit basis for the $\tilde b$05 free directions (Zheng, 21 Jul 2025).

A different algebraic extension uses the ring of $\tilde b$06-linearly correlated fuzzy numbers. If $\tilde b$07 is strongly linearly independent, then $\tilde b$08 is a real vector space isomorphic to $\tilde b$09 via

$\tilde b$10

The $\tilde b$11-cross product $\tilde b$12 turns $\tilde b$13 into a commutative ring, with multiplicative identity $\tilde b$14, and crisp reals embed as $\tilde b$15. In that setting, crisp coefficients act by real scaling under $\tilde b$16, which directly captures the semi-fuzzy case. For the linear fuzzy arithmetic equation

$\tilde b$17

if $\tilde b$18, then the unique solution is

$\tilde b$19

If $\tilde b$20, the equation reduces to a real-scaled fuzzy equation and solvability requires $\tilde b$21 (Laiate et al., 28 Jul 2025).

These developments do not replace the classical SFLS theories based on $\tilde b$22-cuts, block embeddings, H-/M-matrix conditions, or geometric solution sets. They show, rather, that part of the recent SFLS literature is organized around fuzzy-number spaces in which linear algebra, ring operations, Cramer-type formulas, and elimination procedures are available in explicit form. This suggests a broadening of the notion of SFLS from a fixed computational recipe to a family of models whose common feature is the separation between crisp linear structure and fuzzy data, while the solution concept depends on the ambient fuzzy-number space and on whether one seeks a fuzzy vector, a fuzzy set of crisp vectors, or a generalized solution (Zheng, 21 Jul 2025, Laiate et al., 28 Jul 2025).