Linear Equation Systems Under Uncertainty: Applications to Multiproduct Market Equilibrium

Abstract

Market equilibrium models are essential tools within classical economic theory for analyzing the interaction between supply and demand. However, traditional formulations are often based on deterministic relationships and assume the existence of perfect information, an assumption that diverges from real-world conditions, which are characterized by ambiguity and uncertainty. This article addresses the modeling of multiproduct supply and demand equilibrium under uncertainty, using systems of linear equations with fuzzy coefficients and/or variables. By applying fuzzy set theory, the model incorporates the inherent vagueness of supply and demand functions, enabling a more flexible and realistic representation of market behavior. The proposed methodology involves reformulating the equilibrium conditions through fuzzy arithmetic and examining the existence and nature of fuzzy solutions. The theoretical proposals are illustrated through a simplified real-world case involving a Colombian multinational company, demonstrating their applicability and effectiveness.

1. Introduction

In classical economic theory, market equilibrium models are a fundamental tool for analyzing the interaction between the supply and demand of goods and services [1,2,3]. These models, in their traditional formulation, are built on deterministic equations that assume the existence of complete, precise, and perfectly known information by all involved agents. However, this assumption is often inadequate for representing the complexity of the real world, where available information in frequently incomplete, uncertain, or imprecise. This discrepancy between theoretical models and economic reality has spurred the development of alternative approaches that explicitly incorporate uncertainty into the analysis.

Among the various proposed methodologies, fuzzy set theory, which was introduced by Zadeh in 1965, has proven to be particularly effective for modeling economic phenomena in contexts of ambiguity or vagueness [4]. This theory extends the classical notion of a set by allowing an element to belong to a set with a certain degree of membership, represented by a function that takes values in the interval [0, 1]. This approach enables a more flexible representation of reality, which is particularly useful when precise numerical data are unavailable or do not accurately reflect the perceptions of economic agents.

Addressing this type of situation from a static perspective (such as the one adopted in this work) remains the objective of many researchers from various applied perspectives [5,6,7,8,9,10]. However, it is evident that economics, and particularly finance, requires a dynamic perspective capable of handling uncertainty. This need has led to financial modeling proposals based on fuzzy logic, which have proven very useful for population models, option pricing, and other applications (see, for example, [11,12]).

In this article, we propose a model for supply and demand equilibrium in a multiproduct environment under conditions of uncertainty, using fuzzy linear equation systems as the primary tool. The application focuses on two simple cases involving three interrelated goods. The first case examines the equilibrium prices of coffee, tea, and milk in a Colombian company, based on predictions (affected by uncertainty), and the second case addresses a paper company where market study data did not allow for a clear decision. In both cases, it will be demonstrated that formulating supply and demand functions using LR-fuzzy numbers enables decision-making across different scenarios. This type of fuzzy number, widely used in the literature (e.g., [13,14]), allows for an explicit parameterization of the core and support of uncertain values, facilitating both their graphical representation and algebraic treatment [15].

The main objective of this work is twofold: on the one hand, to demonstrate how fuzzy logic tools can be applied to solve economic equilibrium problems involving multiple products and, on the other, to provide a rigorous economic interpretation of the results obtained in terms of fuzzy equilibrium prices and quantities. Throughout this article, the practical implications of this approach are discussed, as well as its advantages over conventional deterministic models.

The structure of the paper is as follows: In Section 2, the fundamental concepts of fuzzy set theory necessary for the development of the model are presented, including the definition of LR-fuzzy numbers and their operational properties. Subsequently, Section 3 introduces the different types of fuzzy linear systems of equations, and Section 4 proposes the corresponding solution methods, presenting algorithms and examples. In Section 5, the system of equations corresponding to market equilibrium is formulated, incorporating the uncertainties inherent in economic functions. Next, two cases illustrating the proposed methodology are solved, highlighting the main technical challenges and possible economic interpretations. Section 6 offers a reflection on the results and their discussion. Finally, Section 7 presents the conclusions and outlines future research directions regarding the use of fuzzy tools in economics.

2. Notation and Preliminary Results

The use of interval numbers and/or fuzzy numbers allows for the representation of a wide range of situations affected by uncertainty, where the use of probability distributions is not justified [4,13,14]. In this work, we will use LR-fuzzy numbers, which were introduced by Dubois and Prade [13]. They are defined as follows:

Definition 1.

An $LR$-fuzzy number $\tilde{M}={(m^L,m^R;{\delta }^L,{\delta }^R)}_{L,R}$ is a fuzzy number whose membership function has the following form:

$${\mu }_{\tilde{M}}\left(x\right)=\left\{\begin{array}{cc}L\left({\displaystyle \frac{m^L-x}{{\delta }^L}}\right)\hfill & \mathit{si}x<m^L\hfill \\ 1\hfill & \mathit{si}{m}^L\le x\le m^R\hfill \\ R\left({\displaystyle \frac{x-m^R}{{\delta }^R}}\right)\hfill & \mathit{si}x>m^R\hfill \end{array}\right.$$

(1)

where $L,R:[0,+\infty ]\to [0,1]$ are strictly decreasing and upper semicontinuous functions satisfying $L\left(0\right)=R\left(0\right)=1$.

If the support of M is a bounded set, with $m^L-{\delta }^L$ as the infimum and $m^R+{\delta }^R$ as the supremum, then the functions L and R are defined on a $[0,1]$ interval and satisfy $L\left(1\right)=R\left(1\right)=0$. When $L\left(z\right)=R\left(z\right)=max(0,1-z)$, M is called a trapezoidal fuzzy number. Furthermore, if $m^L=m^R$, the fuzzy number is called triangular. The standard notation for trapezoidal and triangular fuzzy numbers is $(m_1-{\delta }^L,m_1,m_2,m_2+{\delta }^R)$ and $(m-{\delta }^L,m,m+{\delta }^R)$, respectively. An operational way to work with fuzzy numbers is through the concept of the $\alpha$-cut [4,13].

Definition 2.

Given a fuzzy set $\tilde{M}$ with a membership function ${\mu }_{\tilde{M}}\left(x\right)$, the α-cut of $\tilde{M},$ denoted by $M_{\alpha }$, is the set of all elements x in the universe of discourse such that

$$M_{\alpha }=\{x\in X\mid {\mu }_{\tilde{M}}\left(x\right)\ge \alpha \},$$

(2)

where $\alpha \in [0,1]$ is a cut level.

The union of $\alpha$-cuts, where $\tilde{M}={\bigcup }_{\alpha \in [0,1]}{M}_{\alpha }$, allows for the description of fuzzy numbers. Below, we present a result that enables the description of $LR$-fuzzy numbers using functions and the union of intervals.

Lemma 1.

If $f:[0,1]\to [0,+\infty ]$ is a continuous and strictly increasing function, $g:[0,1]\to [0,+\infty ]$ is a continuous and strictly decreasing function, and if $g\left(\alpha \right)-f\left(\alpha \right)>0$$\forall \alpha \in [0,1]$, then

$$\tilde{M}=\bigcup _{\alpha \in [0,1]}[f\left(\alpha \right),g\left(\alpha \right)]$$

(3)

defines an $LR$-fuzzy number. In this case, we say that $\{f,g\}$ uniquely define the fuzzy number $\tilde{M}$.

Proof. 

Let $m^L=f\left(1\right),m^R=g\left(1\right),{\delta }^L=f\left(1\right)-f\left(0\right)$ and ${\delta }^R=g\left(0\right)-g\left(1\right).$ Since f and g are injective, their inverse functions exist on $Imf$ and $Img$, respectively. Therefore, we can construct the reference functions L and R:

$$\begin{array}{cc}L\left(x\right)=\left\{\begin{array}{cc}0,\hfill & x<m^L-{\delta }^L\hfill \\ f^{-1}\left(x\right),\hfill & m^L-{\delta }^L\le x\le m^L,\hfill \\ 1,\hfill & x>m^R\hfill \end{array}\right.\hfill & R\left(x\right)=\left\{\begin{array}{cc}1,\hfill & x<m^R\hfill \\ g^{-1}\left(x\right),\hfill & m^R\le x\le m^R+{\delta }^R\hfill \\ 0,\hfill & x>m^R+{\delta }^R\hfill \end{array}\right.\hfill \end{array}$$

(4)

From (4), we construct the membership function ${\mu }_{\tilde{M}}\left(x\right)=min\{L\left(x\right),R\left(x\right)\}.$ Through direct computation, it can be verified that (3) describes the $LR$-fuzzy number $\tilde{M}={(m^L,m^R,{\delta }^L,{\delta }^R)}_{L,R}$. □

According to [13,16,17], we can establish the following arithmetic for LR-fuzzy numbers expressed through their $\alpha$-cuts.

Definition 3.

We consider the $LR$-fuzzy numbers $\tilde{x}$ and $\tilde{y}$, which were given by the α-cuts $\tilde{x}=\left(x^L\left(\alpha \right),x^R\left(\alpha \right)\right)$; $\tilde{y}=\left(y^L\left(\alpha \right),y^R\left(\alpha \right)\right)$; $\alpha \in [0,1]$; and $\lambda \in \mathbb{R},$

$$\begin{array}{cc}\hfill \tilde{x}=\tilde{y}& \mathit{if}\mathit{and}\mathit{only}\mathit{if}\forall \alpha ,x^L\left(\alpha \right)=y^L\left(\alpha \right),x^R\left(\alpha \right)=y^R\left(\alpha \right).\hfill \\ \hfill \tilde{x}\oplus \tilde{y}& =\left(x^L\left(\alpha \right)+y^L\left(\alpha \right),x^R\left(\alpha \right)+y^R\left(\alpha \right)\right),\alpha \in [0,1].\hfill \\ \hfill \tilde{x}\ominus \tilde{y}& =\left(x^L\left(\alpha \right)-y^R\left(\alpha \right),x^R\left(\alpha \right)-y^L\left(\alpha \right)\right),\alpha \in [0,1].\hfill \\ \hfill \lambda \tilde{x}& =\left\{\begin{array}{c}\left(\lambda x^L\left(\alpha \right),\lambda x^R\left(\alpha \right)\right),\lambda \ge 0\hfill \\ \left(\lambda x^R\left(\alpha \right),\lambda x^L\left(\alpha \right)\right),\lambda <0,\hfill \end{array}\right.\alpha \in [0,1].\hfill \\ \hfill \tilde{x}\otimes \tilde{y}& =\left(x^L\left(\alpha \right)y^L\left(\alpha \right),x^R\left(\alpha \right)y^R\left(\alpha \right)\right),\alpha \in [0,1].\hfill \end{array}$$

(5)

If the numbers are trapezoidal $\tilde{x}=(m_1,n_1;{\alpha }_1,{\beta }_1)$, $\tilde{y}=(m_2,n_2;{\alpha }_2,{\beta }_2)$ or triangular ($m_1=m_2$, $n_1=n_2$), the operations simplify significantly as follows:

$$\begin{array}{cc}\hfill \tilde{x}=\tilde{y}& \mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{m}_1=m_2,n_1=n_2,{\alpha }_1={\alpha }_2,{\beta }_1={\beta }_2.\hfill \\ \hfill \tilde{x}\oplus \tilde{y}& =(m_1+m_2,n_1+n_2;{\alpha }_1+{\alpha }_2,{\beta }_1+{\beta }_2)\hfill \\ \hfill \tilde{x}\ominus \tilde{y}& =(m_1-n_2,n_1-m_2;{\alpha }_1+{\beta }_2,{\beta }_1+{\alpha }_2)\hfill \\ \hfill \lambda \tilde{x}& =\left\{\begin{array}{c}(\lambda m_1,\lambda n_1;\lambda {\alpha }_1,\lambda {\beta }_1),\lambda \ge 0\hfill \\ (\lambda n_1,\lambda m_1;\lambda {\beta }_1,\lambda \alpha 1),\lambda <0.\hfill \end{array}\right.\hfill \end{array}$$

(6)

The product $\tilde{x}\otimes \tilde{y}$ cannot be expressed as simply as the other operations. In fact, Dubois and Prade [13] propose an approximate option when the tolerances ${\alpha }_1,{\alpha }_2,{\beta }_1$, and ${\beta }_2$ are small relative to $|m_1|,|m_2|,|n_1|$, and $|n_2|$. In this case, for non-negative trapezoidal numbers, we have

$$\tilde{x}\otimes \tilde{y}\approx (m_1{m}_2,n_1{n}_2;m_1{\alpha }_2+m_2{\alpha }_1,m_1{\beta }_2+m_2{\beta }_1),m_1,m_2,n_1,n_2\ge 0.$$

(7)

If they are triangular numbers ($m_1=m_2,n_1=n_2$), this can be extended to negative fuzzy numbers, where

$$\tilde{x}\otimes \tilde{y}\approx (m_1{m}_2;|m_1|{\alpha }_2+|m_2|{\alpha }_1,|m_1|{\beta }_2+|m_2\left|{\beta }_1\right).$$

(8)

As we will demonstrate in this work, the arithmetic of $\alpha$-cuts enables the resolution of systems of linear equations.

Remark 1.

In this work, we will use triangular and trapezoidal numbers to model uncertainty. However, it is necessary to work within the framework of $LR$-fuzzy numbers because when solving the systems, the membership functions of the solutions are not piecewise linear. This is due to the fact that, when the coefficients, variables, and/or right-hand side are parameterized using α-cuts, the computation of the solutions involves products of parameters, resulting in a loss of linearity.

3. Problem Formulation

In this work, we are interested in market equilibrium models where the supply and demand functions admit a linear formulation and depend on the prices of multiple products. This situation can be formulated, at least approximately, using systems of linear equations.

As noted in the introduction, supply and demand functions may be affected by uncertainty in their coefficients and even in their variables. Consequently, we encounter fuzzy linear equations systems. In the extensive literature addressing these systems [18,19], it is common to consider the unknowns as LR-fuzzy numbers to be calculated [18]. In the context of linear algebra, it is less common to assume that the variables need not be fuzzy numbers or even fuzzy sets [4]. However, this approach is widely used in linear programming [20]. In this work, as shown in

Table 1. Types of fuzzy systems.

	Formulation	Source of Uncertainty	Auxiliary Parametric System
(FC)	$\tilde{A}x=\tilde{b}$	In the coefficients and independent terms.	$A\left(\alpha \right)·x=b\left(\alpha \right)$
(FV)	$A\tilde{x}=\tilde{b}$	In the variables and independent terms.	$A·x\left(\alpha \right)=b\left(\alpha \right)$
(FLS)	$\tilde{A}\otimes \tilde{x}=\tilde{b}$	In the coefficients, variables, and independent terms.	$A\left(\alpha \right)·x\left(\alpha \right)=b\left(\alpha \right)$

FC = Fuzzy Coefficient, FV = Fuzzy Variable, and FLS = Fuzzy Linear System.

, we will address both approaches.

In columns 2 and 3 of Table 1, the different types of systems are shown, and in column 4, the auxiliary crisp parametric systems that enable the resolution of systems affected by uncertainty are presented. Note that all the cases mentioned in Table 1 use fuzzy arithmetic because if even a single quantity is fuzzy, this turns the equations into fuzzy equations.

For a vector or set of vectors to be a solution to the systems described in Table 1, different types of requirements must be established.

Definition 4.

The set $X=\left\{\left(x_1\left(\alpha \right),x_2\left(\alpha \right),\cdots ,x_n\left(\alpha \right)\right),\alpha \in [0,1]\right\}$ is a solution to (FC) if and only if, for any $\alpha \in [0,1],x\left(\alpha \right)=\left(x_1\left(\alpha \right),x_2\left(\alpha \right),\cdots ,x_n\left(\alpha \right)\right)$ is a solution to $A·x\left(\alpha \right)=b\left(\alpha \right)$.

However, solutions obtained from the associated parametric systems cannot be directly considered solutions to FVs and the FLS, as the unknowns must be fuzzy numbers. Thus, according to [21], we define the fuzzy solution (Table 1) as follows.

Definition 5.

Let $X=\left\{(x_i^L\left(\alpha \right),x_i^R\left(\alpha \right)),\alpha \in [0,1],1\le i\le n\right\}$ be the solution to $Ax\left(\alpha \right)=b\left(\alpha \right)$ or $A\left(\alpha \right)x\left(\alpha \right)=b\left(\alpha \right)$ (see Table 1). For each $i\in \{1,2,\cdots ,n\},$ we calculate

$$u_i^L\left(\alpha \right)=min\left\{x_i^L\left(\alpha \right),x_i^R\left(\alpha \right)\right\},u_i^R\left(\alpha \right)=max\left\{x_i^L\left(\alpha \right),x_i^R\left(\alpha \right)\right\}.$$

(9)

If $\left(x_i^L\left(\alpha \right),x_i^R\left(\alpha \right)\right),\alpha \in [0,1],$ determines a fuzzy number, $1\le i\le n$, then

$$U=\left\{(u_i^L\left(\alpha \right),u_i^R\left(\alpha \right)),\alpha \in [0,1],1\le i\le n\right\}$$

(10)

is a fuzzy solution to the system $A\tilde{x}=\tilde{b}$ or $\tilde{A}\otimes \tilde{x}=\tilde{b}$.

When (10) does not define fuzzy numbers, we can extend the proposal by Friedman, Ming, and Kandel [21] to obtain an approximate (weak) solution:

(i)

If ${u_i^L}^{\ast }\left(\alpha \right)$ is not monotonically increasing, we can construct the following alternative function:

$$u_i^{L\ast }\left(\alpha \right)=\left\{\begin{array}{cc}{u}_i^L\left(1\right),\hfill & \mathrm{si}{u}_i^L\left(0\right)\ge u_i^L\left(1\right),\hfill \\ u_i^L\left(0\right)+\alpha (u_i^L\left(1\right)-u_i^L\left(0\right)),\hfill & \mathrm{si}{u}_i^L\left(0\right)<u_i^L\left(1\right).\hfill \end{array}\right.$$

(11)

(ii)

If $u_i^{R\ast }\left(\alpha \right)$ is not monotonically decreasing, we can construct the following alternative function:

$$u_i^{R\ast }\left(\alpha \right)=\left\{\begin{array}{cc}{u}_i^R\left(1\right),\hfill & \mathrm{si}{u}_i^R\left(0\right)\le u_i^R\left(1\right),\hfill \\ u_i^R\left(0\right)+\alpha (u_i^R\left(0\right)-u_i^R\left(1\right)),\hfill & \mathrm{si}{u}_i^R\left(0\right)>u_i^R\left(1\right).\hfill \end{array}\right.$$

(12)

According to (9), $u_i^L\left(1\right)\le u_i^R\left(1\right)$; then,

$$U^{\ast }=\left\{(u_i^{L\ast }\left(\alpha \right),u_i^{R\ast }\left(\alpha \right)),\alpha \in [0,1],1\le i\le n\right\}$$

(13)

is a weak fuzzy solution of $A\tilde{x}=\tilde{b}$ or $\tilde{A}\otimes \tilde{x}=\tilde{b}$.

In the extreme case of weak fuzzy solutions, there are degenerate solutions formed by crisp numbers.

Theorem 1.

Consider the FLS system (Table 1) with ${\tilde{a}}_{ij}={(a_{ij}^L,a_{ij}^R;{\alpha }_{ij},{\beta }_{ij})}_{LR}$, ${\tilde{x}}_j={(x_j^L,x_j^R;{\gamma }_j,{\delta }_j)}_{LR}$, and ${\tilde{b}}_i={(b_i^L,b_i^R;p_i,p_i)}_{LR},$ $1\le i,j\le n$. If $x^{\ast }=(x_1^{L\ast },\cdots ,x_n^{L\ast },$$x_1^{R\ast },\cdots ,x_n^{R\ast })$ is a solution to

$$\left.\begin{array}{ccc}& \sum _{j=1}^n{a}_{ij}^L{x}_j^L=b_i^L,& 1\le i\le n\\ & \sum _{j=1}^n{a}_{ij}^R{x}_j^R=b_i^R,& 1\le i\le n\end{array}\right\}$$

then

$${\tilde{x}}_j=(x_j^{L\ast },x_j^{R\ast };0,0)$$

(14)

is a weak solution, which we will call a degenerate fuzzy solution to $A\tilde{x}=\tilde{b}$ or $\tilde{A}\otimes \tilde{x}=\tilde{b}$.

In the following section, we will present methods to obtain solutions for the fuzzy systems described in Table 1.

4. Resolution Methods

4.1. Systems with Precise Unknowns

In 1986, Carlsson and Korhonen, working with linear programming models, proposed in [20] the model’s uncertain coefficients using intervals, such as $\{a,b\}$. In these intervals, the first number, a, represents the risk-free value, and the second, b, represents the impossible value, i.e.,

$${\mu }_C\left(a\right)=1,{\mu }_C\left(b\right)=0.$$

(15)

Note that we do not write $[a,b]$ because it is not necessarily true that $a\le b$. With this notation, we can incorporate uncertainty into the parameters of a system of linear equations as follows:

$$\left.\begin{array}{ccc}\{a_{11}^L,a_{11}^R\}{x}_1+\{a_{12}^L,a_{12}^R\}{x}_2+\cdots +\{a_{1n}^L,a_{1n}^R\}{x}_n& =& \{b_1^L,b_1^R\}\\ \{a_{21}^L,a_{21}^R\}{x}_1+\{a_{22}^L,a_{22}^R\}{x}_2+\cdots +\{a_{2n}^L,a_{2n}^R\}{x}_n& =& \{b_2^L,b_2^R\}\\ ⋮& & ⋮\\ \{a_{m1}^L,a_{m1}^R\}{x}_1+\{a_{m2}^L,a_{m2}^R\}{x}_2+\cdots +\{a_{m1}^L,a_{m1}^R\}{x}_n& =& \{b_m^L,b_m^R\}\end{array}\right\}$$

(16)

For simplicity, we assume that uncertainty can be modeled with linear functions. In this case, we would have the following membership functions:

$$\begin{array}{cc}\begin{array}{c}\hfill {\mu }_{a_{ij}}\left(x\right)\\ \hfill (a_{ij}^L<a_{ij}^R)\end{array}=\left\{\begin{array}{cc}1\hfill & x<a_{ij}^L\hfill \\ {\displaystyle \frac{x-a_{ij}^R}{a_{ij}^L-a_{ij}^R}}\hfill & a_{ij}^L\le x\le a_{ij}^R,\hfill \\ 0\hfill & x>a_{ij}^R\hfill \end{array}\right.& \begin{array}{c}\hfill {\mu }_{a_{ij}}\left(x\right)\\ \hfill (a_{ij}^L>a_{ij}^R)\end{array}=\left\{\begin{array}{cc}0\hfill & x<a_{ij}^R\hfill \\ {\displaystyle \frac{x-a_{ij}^R}{a_{ij}^L-a_{ij}^R}}\hfill & a_{ij}^R\le x\le a_{ij}^L\hfill \\ 1\hfill & x>a_{ij}^L\hfill \end{array}\right.\end{array}$$

(17)

$$\begin{array}{cc}\begin{array}{c}\hfill {\mu }_{b_i}\left(x\right)\\ \hfill (b_i^L<b_i^R)\end{array}=\left\{\begin{array}{cc}1\hfill & x<b_i^L\hfill \\ {\displaystyle \frac{x-b_i^R}{b_i^L-b_i^R}}\hfill & b_i^L\le x\le b_i^R,\hfill \\ 0\hfill & x>b_i^R\hfill \end{array}\right.& \begin{array}{c}\hfill {\mu }_{b_i}\left(x\right)\\ \hfill (b_i^L>b_i^R)\end{array}=\left\{\begin{array}{cc}0\hfill & x<b_i^R\hfill \\ {\displaystyle \frac{x-b_i^R}{b_i^L-b_i^R}}\hfill & b_i^R\le x\le b_i^L\hfill \\ 1\hfill & x>b_i^L\hfill \end{array}\right.\end{array}$$

(18)

From (17) and (18), we can express the coefficients of (16) as follows:

$$a_{ij}=(a_{ij}^L-a_{ij}^R){\mu }_{a_{ij}}+a_{ij}^R,b_j=(b_j^L-b_j^R){\mu }_{b_j}+b_j^R,{\mu }_{a_{ij}},{\mu }_{b_j}\in [0,1].$$

(19)

Obviously, introducing $m(n+1)$ parameters into System (16) via (19) adds computational complexity. A practical way to avoid this issue is to adopt the most pessimistic scenario regarding the plausibility of all coefficients [20], i.e.,

$$\alpha =\underset{i,j}{min}\{{\mu }_{a_{ij}},{\mu }_{b_j}\}.$$

(20)

With this, the solution to System (16) can be obtained through an auxiliary crisp system of linear equations:

$$A\left(\alpha \right)X=B\left(\alpha \right).$$

(21)

The usual way to provide the solution to the decision-maker is by discretizing $\alpha$, for instance, ${\alpha }_k=k/10,0\le k\le 10,$

$$x_1\left({\alpha }_k\right),x_1\left({\alpha }_k\right),\cdots x_1\left({\alpha }_k\right),0\le k\le 10.$$

(22)

In practice, the described method is very useful when starting from a system of linear equations that has no solution or whose solution does not satisfy certain requirements [22,23], as we will see in the applications section. Below, we present a result along these lines.

Theorem 2.

Given an incompatible linear system $Ax=b$, where $A=\left[a_{ij}\right]$ is a square matrix of order n, if there exists a minor M of order $n-1$ that is non-zero, then it is possible to construct a compatible determined system $A^{\prime }x=b$, where $A^{\prime }=\left[a_{ij}^{\prime }\right]$, with $a_{ij}^{\prime }=a_{ij}$, except for the term $a_{i_0{j}_0}^{\prime }=a_{i_0{j}_0}+ϵ$, where $ϵ\ne 0$.

Proof. 

Without loss of generality, we can assume that $M=\left[a_{ij}\right],2\le i,j,\le n$, and we know that $detM\ne 0$. Then, given $ϵ\ne 0$ we set

$$a_{ij}^{\prime }=\left\{\begin{array}{cc}{a}_{11}+ϵ\hfill & \mathrm{si}(i,j)=(1,1)\hfill \\ a_{ij}\hfill & \mathrm{si}(i,j)\ne (1,1)\hfill \end{array}\right.$$

(23)

Given the matrix $A^{\prime }=\left[a_{ij}^{\prime }\right]$, we observe that $det{A}^{\prime }=ϵdetM\ne 0$ through direct computation. Therefore, $A^{\prime }x=b$ is a compatible determined system. □

Of course, avoiding infeasibility (to the extent possible) is not the only utility of this method; it is particularly useful when experts or the decision-maker lack sufficient information to describe a membership function but can establish intervals for certain coefficients [20,22].

4.2. Systems with Fuzzy Unknowns

In this section, we address systems where the unknowns can be formulated as fuzzy numbers. According to Table 1, these correspond to two cases: $A\tilde{x}=\tilde{b}$ and $\tilde{A}\otimes \tilde{x}=\tilde{b}.$ As will be seen in the following two subsections, both cases follow a similar reasoning, although the crisp auxiliary systems and the elements of the system described as $\alpha$-cuts are different. Figure 1 shows a common flow diagram for both methods.

In the following two subsections, we present the details of these two algorithms, illustrating their operational procedures.

4.2.1. Systems of the Type $A\tilde{x}=\tilde{b}$

In 1986, Friedman, Ming, and Kandel [21] proposed a method that is quite practical for solving this type of systems, although it may sometimes present difficulties [24,25]. Expressed algorithmically, the proposal can be described as follows:

STEP 1:

Identify the problem whose mathematical representation is a system of fuzzy linear equations, where

$$A\tilde{x}=\tilde{b}.$$

(24)

STEP 2:

Descompose each fuzzy number into $\alpha$-cuts, where

$$\widetilde{x_j}=\bigcup _{\alpha \in [0,1]}[x_j^L\left(\alpha \right),x_j^R\left(\alpha \right)],\widetilde{b_j}=\bigcup _{\alpha \in [0,1]}[b_j^L\left(\alpha \right),b_j^R\left(\alpha \right)],1\le i,j\le n.$$

(25)

STEP 3:

Construct a new auxiliary crisp system of $2n$ equations and $2n$ unknowns as a function of the parameter $\alpha \in [0,1]$ as follows:

$$S·x\left(\alpha \right)=b\left(\alpha \right),\alpha \in [0,1],$$

(26)

where the matrix S is constructed as

$$\begin{array}{ccc}{s}_{ij}=s_{i+n,j+n}=\left\{\begin{array}{cc}{a}_{ij}\hfill & \mathrm{si}{a}_{ij}\ge 0\hfill \\ 0\hfill & \mathrm{si}{a}_{ij}<0\hfill \end{array}\right.\hfill & s_{i,j+n}=s_{i+n,j}=\left\{\begin{array}{cc}-a_{ij}\hfill & \mathrm{si}{a}_{ij}\le 0\hfill \\ 0\hfill & \mathrm{si}{a}_{ij}<0\hfill \end{array}\right.\hfill & 1\le i,j\le n,\hfill \end{array}$$

(27)

where the matrix $x\left(\alpha \right)={\left(x_1^L\left(\alpha \right),\cdots ,x_n^L\left(\alpha \right),-x_1^R\left(\alpha \right),\cdots ,-x_n^R\left(\alpha \right)\right)}^T$ and the matrix $b\left(\alpha \right)={\left(b_1^L\left(\alpha \right),\cdots ,b_n^L\left(\alpha \right),-b_1^R\left(\alpha \right),\cdots ,-b_n^R\left(\alpha \right)\right)}^T$.

STEP 4:

Solve the auxiliary crisp system generates in Step 3 to obtain $x_j^{\ast L}\left(\alpha \right)$ y $x_j^{\ast R}\left(\alpha \right)$, $1\le j\le n.$

STEP 5:

Calculate $u_j^L\left(\alpha \right)=min\left\{x_j^{\ast L}\left(\alpha \right),x_j^{\ast R}\left(\alpha \right)\right\},u_j^R\left(\alpha \right)=max\left\{x_j^{\ast L}\left(\alpha \right),x_j^{\ast R}\left(\alpha \right)\right\}.$

STEP 6:

If $\left(u_j^L\left(\alpha \right),u_j^R\left(\alpha \right)\right),\alpha \in [0,1],$ determines a fuzzy number, where $1\le j\le n$, then

$$U=\left\{(u_i^L\left(\alpha \right),u_i^R\left(\alpha \right)),\alpha \in [0,1],1\le i\le n\right\}$$

(28)

is a fuzzy solution to the system $A\tilde{x}=\tilde{b}$.

Below, we apply the algorithm to an example where the matrix of independent terms $\tilde{b}$ consists of triangular fuzzy numbers.

Example 1.

Find the solutions to the following system of equations:

$$\left.\begin{array}{ccc}\hfill -2\tilde{x}+3\tilde{y}& =& (0,1,2)\hfill \\ \hfill 2\tilde{x}-2\tilde{y}& =& (-4,-2,-1)\hfill \end{array}\right\}$$

(29)

According to Step 3, we solve the following auxiliary system:

$$\left(\begin{array}{cccc}0& 3& 2& 0\\ 2& 0& 0& 2\\ 2& 0& 0& 3\\ 0& 2& 2& 0\end{array}\right)\left(\begin{array}{c}{x}^L\left(\alpha \right)\\ y^L\left(\alpha \right)\\ -x^R\left(\alpha \right)\\ -y^R\left(\alpha \right)\end{array}\right)=\left(\begin{array}{c}\alpha \\ -4+2\alpha \\ -2+\alpha \\ 1+\alpha \end{array}\right)$$

(30)

The solution to the parametric problem is $x^L\left(\alpha \right)=-4+2\alpha ,x^R\left(\alpha \right)={\displaystyle \frac{-\alpha -3}{2}},$ $y^L\left(\alpha \right)=\alpha -2,y^R\left(\alpha \right)=-1,\alpha \in [0,1].$ Therefore, $\tilde{x}=(-4,-2,-1.5),$ $\tilde{y}=(-2,-1,-1)$ is the solution of the fuzzy system.

4.2.2. Systems of the Type $\tilde{A}\otimes \tilde{x}=\tilde{b}$

In this section, we present a method for solving this type of system based on the works by Pandit [26,27], Nasseri et al. [17], and Friedman et al. [21], among others. Below, we present an algorithm that may prove to be operational.

STEP 1:

Identify the problem whose mathematical representation is a system of fuzzy linear equations.

$$\tilde{A}\otimes \tilde{x}=\tilde{b}$$

(31)

STEP 2:

Decompose each fuzzy number into $\alpha$-cuts, where

$$\begin{array}{cc}\widetilde{a_{ij}}=\bigcup _{\alpha \in [0,1]}[a_{ij}^L\left(\alpha \right),a_{ij}^R\left(\alpha \right)],\hfill & \widetilde{x_j}=\bigcup _{\alpha \in [0,1]}[x_j^L\left(\alpha \right),x_j^R\left(\alpha \right)],\hfill \\ \widetilde{b_j}=\bigcup _{\alpha \in [0,1]}[b_j^L\left(\alpha \right),b_j^R\left(\alpha \right)],\hfill & 1\le i,j\le n.\hfill \end{array}$$

(32)

STEP 3:

Construct a new auxiliary crisp system, $S\left(\alpha \right)x\left(\alpha \right)=b\left(\alpha \right)$, of $2n$ equations, and $2n$ unknowns as a function of the parameter $\alpha \in [0,1]$ as follows:

$$\left(\begin{array}{cccccc}{a}_{11}^L\left(\alpha \right)& \cdots & a_{1n}^L\left(\alpha \right)& 0& \cdots & 0\\ a_{21}^L\left(\alpha \right)& \cdots & a_{2n}^L\left(\alpha \right)& 0& \cdots & 0\\ & ⋮& & & ⋮\\ a_{n1}^L\left(\alpha \right)& \cdots & a_{nn}^L\left(\alpha \right)& 0& \cdots & 0\\ 0& \cdots & 0& a_{11}^R\left(\alpha \right)& \cdots & a_{ij}^R\left(\alpha \right)\\ 0& \cdots & 0& a_{21}^R\left(\alpha \right)& \cdots & a_{2n}^R\left(\alpha \right)\\ & ⋮& & & ⋮\\ 0& \cdots & 0& a_{n1}^R\left(\alpha \right)& \cdots & a_{nn}^R\left(\alpha \right)\end{array}\right)\left(\begin{array}{c}{x}_1^L\left(\alpha \right)\\ x_2^L\left(\alpha \right)\\ ⋮\\ x_n^L\left(\alpha \right)\\ x_1^R\left(\alpha \right)\\ x_2^R\left(\alpha \right)\\ ⋮\\ x_n^R\left(\alpha \right)\end{array}\right)=\left(\begin{array}{c}{b}_1^L\left(\alpha \right)\\ b_2^L\left(\alpha \right)\\ ⋮\\ b_n^L\left(\alpha \right)\\ b_1^R\left(\alpha \right)\\ b_2^R\left(\alpha \right)\\ ⋮\\ b_n^R\left(\alpha \right)\end{array}\right)$$

(33)

STEP 4:

For negative fuzzy numbers in the matrix $S\left(\alpha \right)$, swap the values $a_{ij}^L\left(\alpha \right)$ con $a_{ij}^R\left(\alpha \right)$.

STEP 5:

Calculate the solution to the crisp system generated in the previous step, namely $x_j^{\ast L}\left(\alpha \right)$ and $x_j^{\ast R}\left(\alpha \right)$, where $1\le j\le n.$

STEP 6:

Calculate $u_j^L\left(\alpha \right)=min\left\{x_j^{\ast L}\left(\alpha \right),x_j^{\ast R}\left(\alpha \right)\right\},u_j^R\left(\alpha \right)=max\left\{x_j^{\ast L}\left(\alpha \right),x_j^{\ast R}\left(\alpha \right)\right\}.$

STEP 7:

If $\left(u_j^L\left(\alpha \right),u_j^R\left(\alpha \right)\right),\alpha \in [0,1],$ determines a fuzzy number, where $1\le j\le n$, then

$$U=\left\{(u_i^L\left(\alpha \right),u_i^R\left(\alpha \right)),\alpha \in [0,1],1\le i\le n\right\}$$

(34)

is a fuzzy solution to the system $\tilde{A}\left(\alpha \right)\otimes \tilde{x}=\tilde{b}$.

In [26,27], a sufficient condition is analyzed for the $\alpha$-cuts of (34) to define a vector of fuzzy numbers that is a solution to (31).

Theorem 3. [Pandit’s condition]:

The fuzzy solution to System (31) exists if

$$\forall \alpha ,\beta \in [0,1],\alpha \le \beta ,x_j^{\ast L}\left(\alpha \right)\le x_j^{\ast L}\left(\beta \right)\le x_j^{\ast R}\left(\beta \right)\le x_j^{\ast R}\left(\alpha \right),1\le j\le n.$$

(35)

Pandit’s condition is less restrictive than Definition 4, as it queue admits so-called weak fuzzy solutions (see (13)).

Remark 2.

If the inequalities in (35) hold strictly, the functions $x^L\left(\alpha \right)$ and $x^R\left(\alpha \right)$ satisfy the conditions of Lemma 1, as they are continuous throughout their domain (being polynomial or rationals) since they are solutions to linear systems.

Below, we apply the previous algorithm to an example where the uncertainty is modeled with triangular fuzzy numbers.

Example 2.

Find the solution to the following system:

$$\left.\begin{array}{ccc}\hfill (-3,-2,-1)\otimes \tilde{x}+(1,3,4)\otimes \tilde{y}& =& (0,1,2)\hfill \\ \hfill (1,2,3)\otimes \tilde{x}+(-3,-2,0)\otimes \tilde{y}& =& (-4,-2,-1)\hfill \end{array}\right\}$$

According to (33), in Step 3, we express the system as

$$\left(\begin{array}{cccc}\alpha -3& 2\alpha +1& 0& 0\\ \alpha +1& \alpha -3& 0& 0\\ 0& 0& -\alpha -1& 4-\alpha \\ 0& 0& 3-\alpha & -2\alpha \end{array}\right)\left(\begin{array}{c}{x}^L\left(\alpha \right)\\ y^L\left(\alpha \right)\\ x^R\left(\alpha \right)\\ y^R\left(\alpha \right)\end{array}\right)=\left(\begin{array}{c}\alpha \\ -4+2\alpha \\ 2-\alpha \\ -1-\alpha \end{array}\right)$$

(36)

Considering Step 4 of the algorithm and since (36) shows that the numbers $s_{11}=\alpha -3$ and $s_{22}=\alpha -3$ are negative, they are swapped with $s_{33}$ and $s_{44}$, respectively.

$$\left(\begin{array}{cccc}-\alpha -1& 2\alpha +1& 0& 0\\ \alpha +1& -2\alpha & 0& 0\\ 0& 0& \alpha -3& 4-\alpha \\ 0& 0& 3-\alpha & \alpha -3\end{array}\right)\left(\begin{array}{c}{x}^L\left(\alpha \right)\\ y^L\left(\alpha \right)\\ x^R\left(\alpha \right)\\ y^R\left(\alpha \right)\end{array}\right)=\left(\begin{array}{c}\alpha \\ -4+2\alpha \\ 2-\alpha \\ -1-\alpha \end{array}\right)$$

(37)

The parametric solution, $\alpha \in [0,1]$, is

$$\begin{array}{cc}{\displaystyle x^L\left(\alpha \right)=\frac{-6{\alpha }^2+6\alpha +4}{-\alpha -1},}\hfill & {\displaystyle x^R\left(\alpha \right)=\frac{-2{\alpha }^2+8\alpha -2}{\alpha -3},}\hfill \\ {\displaystyle y^L\left(\alpha \right)=3\alpha -4,}\hfill & {\displaystyle y^R\left(\alpha \right)=1-2\alpha .}\hfill \end{array}$$

(38)

In Figure 2, the curves $L\left(x\right)$ and $R\left(x\right)$ (left figure) and $L\left(y\right)$ and $R\left(y\right)$ (right figure) are shown; they were obtained from Solution (38). We can verify that $x^L\left(\alpha \right)$ and $x^R\left(\alpha \right)$ do not define a fuzzy number. According to Theorem 3, for $\tilde{x},\tilde{y}$ to be solutions of the FSEL (29), the following condition must hold: $\forall \alpha ,\beta \in [0,1],\alpha \le \beta ,$ $x^L\left(\alpha \right)\le x^L\left(\beta \right)\le x^R\left(\beta \right)\le x^R\left(\alpha \right).$ If we take $\alpha =0.05$ and $\beta =0.15$, the condition is not satisfied because $x^L\left(0.05\right)\approx -4.081>x^L\left(0.15\right)=-4.143.$ In the graph of $\tilde{x}$ in Figure 2, we can observe that $x^L\left(\alpha \right)$ is not a decreasing function. In fact, the function has a relative minimum at $\alpha =\sqrt{4/3}-1.$ If we apply (9), since $x^L\left(0\right)<x^L\left(1\right)$, we construct $x^{L\ast }=x\left(0\right)+\alpha (x^L\left(1\right)-x^L\left(0\right)),\alpha \in [0,1]$, obtaining $L^{\ast }\left(x\right)$ (see Figure 2).

For $\tilde{x}=(x^{L\ast }\left(\alpha \right),x^R\left(\alpha \right)),\tilde{y}=(y^L\left(\alpha \right),y^R\left(\alpha \right)),\alpha \in [0,1]$, the weak fuzzy solution of the system is given by the following triangular fuzzy numbers:

$$\tilde{x}=\left(-4,-2,{\displaystyle \frac{2}{3}}\right),\tilde{y}=(-4,-1,1).$$

(39)

Remark 3.

Note that, as pointed out in Remark 1, the solution of the system given in (38) is not linear, as can be seen in Figure 2.

Theorem 4.

Let $A\tilde{x}=\tilde{b}$ in a system, where $A\in M_{n\times n}\left(\mathbb{R}\right)$. If A is non-singular, then the system has the solution $\tilde{x}$.

Proof. 

We consider each element $a_{ij}$ of the matrix A as a trivial fuzzy number, expressed in terms of $\alpha$-cuts as $a_{ij}\left(\alpha \right)=(a_{ij},a_{ij}).$ Taking into account (33), we construct the matrix $S={\left[s_{ij}\right]}_{2n\times 2n},$ which is given by

$$s_{ij}=a_{ij},s_{n+i,j}=0,s_{i,n+j}=0,s_{n+i,n+j}=a_{ij},1\le i,j\le n.$$

(40)

By construction, $detS={(detA)}^2>0,$ so $S^{-1}$ exists. We define

$$\left(\begin{array}{c}{x}_1^L\left(\alpha \right)\\ ⋮\\ x_n^L\left(\alpha \right)\\ x_1^R\left(\alpha \right)\\ ⋮\\ x_n^R\left(\alpha \right)\end{array}\right)=S^{-1}\left(\begin{array}{c}{b}_1^L\left(\alpha \right)\\ ⋮\\ b_n^L\left(\alpha \right)\\ b_1^R\left(\alpha \right)\\ ⋮\\ b_n^R\left(\alpha \right)\end{array}\right).$$

(41)

According to (9), we calculate the values $u_i^L\left(\alpha \right)=min\{x_i^L\left(\alpha \right),x_i^R\left(\alpha \right)\}$ and $u_i^R\left(\alpha \right)=max\{x_i^L\left(\alpha \right),x_i^R\left(\alpha \right)\}$, where $1\le i\le n.$ Then, given the fuzzy numbers ${\tilde{u}}_i=(u_i^L\left(\alpha \right),u_i^R\left(\alpha \right)),1\le i\le n$, it follows that $\tilde{u}=({\tilde{u}}_1,{\tilde{u}}_2,\cdots ,{\tilde{u}}_n)$ is a solution to the system. □

Remark 4.

Note that the algorithms in Section 4.2.1 and Section 4.2.2 are not equivalent when the coefficients are crisp. For example, we consider

$$\left.\begin{array}{ccc}\hfill \tilde{x}\oplus \tilde{y}& =& \tilde{3}\hfill \\ \hfill \tilde{x}\ominus \tilde{y}& =& \tilde{1}\hfill \end{array}\right\}.$$

(42)

If we model the fuzzy numbers $\tilde{3}$ and $\tilde{1}$ using the triangular fuzzy numbers $\tilde{3}=(2,3,5)$ and $\tilde{1}=(1,1,2)$ with the method described in Section 4.2.1, the system (42) has no solution because the matrix S associated with the system (see (27)) is singular. However, with the method given in Section 4.2.2, the solution is $\tilde{x}=\left({\displaystyle \frac{3}{2}},2,{\displaystyle \frac{7}{2}}\right),\tilde{y}=\left({\displaystyle \frac{1}{2}},1,{\displaystyle \frac{3}{2}}\right).$

4.3. With Small Tolerances

In the literature dedicated to the resolution of fuzzy systems [9,16,17,18,21,24,26,27,28,29,30,31,32], not enough attention is usually paid to the magnitude of the tolerances. When working with $LR$-fuzzy numbers $\tilde{M}={(m;{\delta }^L,{\delta }^R)}_{L,R}$ in which ${\delta }^L\ll \left|m\right|$ and ${\delta }^R\ll \left|m\right|$, we can use the arithmetic described in (8) to solve the systems.

If we model the system $\tilde{A}\otimes \tilde{x}=\tilde{b}$ with $LR$-fuzzy numbers ${\tilde{a}}_{ij}=(a_{ij};{\alpha }_{ij},{\beta }_{ij})$, where ${\alpha }_{ij},{\beta }_{ij}\ll \left|a_{ij}\right|$, ${\tilde{x}}_j=(x_j;{\gamma }_j,{\delta }_j)$, and ${\tilde{b}}_i=(b_i;p_i,q_i)$, $1\le i,j\le n$, and taking into account the arithmetic described in (8), the crisp system to be solved is the following:

$$\left.\begin{array}{ccc}\left(\mathrm{CFES}\right)\hfill & \sum _{j=1}^n{a}_{ij}{x}_j=b_i,\hfill & 1\le i\le n\hfill \\ \hfill & \sum _{j=1}^n\left(|a_{ij}|{\gamma }_j+\left|x_j\right|{\alpha }_{ij}\right)=p_i,\hfill & 1\le i\le n\hfill \\ \hfill & \sum _{j=1}^n\left(|a_{ij}|{\delta }_j+\left|x_j\right|{\beta }_{ij}\right)=q_i,\hfill & 1\le i\le n\hfill \end{array}\right\}$$

(43)

Theorem 5.

If the elements $a_{ij}$, $1\le i,j,\le n$ of $\tilde{A}$ are non-negative, then (43) has a solution if and only if $det\left[a_{ij}\right]\ne 0.$

Proof. 

Since $det\left[a_{ij}\right]\ne 0,$${\sum }_{j=1}^n{a}_{ij}{x}_j=b_i,1\le i\le n,$ has a solution $(x_1^{\ast },x_2^{\ast },\cdots ,x_n^{\ast })$. Additionally, since $|a_{ij}|=a_{ij},$ the systems

$$\sum _{j=1}^n{a}_{ij}{\gamma }_j=p_i-\sum _{j=1}^n|x_j^{\ast }|{\alpha }_{ij},\sum _{j=1}^n{a}_{ij}{\delta }_j=q_i-\sum _{j=1}^n\left|x_j^{\ast }\right|{\beta }_{ij},1\le i\le n,$$

(44)

also have solutions $({\gamma }_1^{\ast },{\gamma }_2^{\ast },\dots ,{\gamma }_n^{\ast })$ and $({\delta }_1^{\ast },{\delta }_2^{\ast },\dots ,{\delta }_n^{\ast })$, respectively. □

However, Theorem 5 does not guarantee the existence of a solution to $\tilde{A}\otimes \tilde{x}=\tilde{b}$. It is necessary to impose non-negativity on ${\gamma }_j^{\ast },{\delta }_j^{\ast },$$1\le j\le n.$

Theorem 6.

Given $x_j^{\ast },{\gamma }_j^{\ast },{\delta }_j^{\ast },$$1\le j\le n,$ as a solution to (43), if ${\gamma }_j,{\delta }_j\ge 0,1\le j\le n$, then the vector

$$({\tilde{x}}_1^{\ast },{\tilde{x}}_2^{\ast },\cdots ,{\tilde{x}}_n^{\ast }),$$

(45)

with ${\tilde{x}}_j=(x_j^{\ast };{\gamma }_j^{\ast },{\delta }_j^{\ast }),1\le j\le n,$ is a fuzzy solution to the system $\tilde{A}\otimes \tilde{x}=\tilde{b}$ (see Definition 4).

Proof. 

We define $L\left(z\right)=R\left(z\right)=max\{0,1-z\}$ by applying Lemma 1, so ${\tilde{x}}_j=(x_j^{\ast };{\gamma }_j^{\ast },{\delta }_j^{\ast }),$$1\le j\le n,$ is an $LR$-fuzzy number, thus satisfying Definition 4. □

Next, we present an example applying (43) and Theorem 6 where all elements of the system are modeled as triangular fuzzy numbers.

Example 3.

Solve the following system of equations:

$$\left.\begin{array}{ccc}\hfill (-10;0.25,1)\otimes \widetilde{x_1}+(15;0.5,0)\otimes \widetilde{x_2}& =& (70;11,20)\hfill \\ \hfill (11;0.5,0.6)\otimes \widetilde{x_1}+(20;0.25,0.25)\otimes \widetilde{x_2}& =& (69;12,26)\hfill \end{array}\right\}$$

(46)

Let ${\tilde{x}}_i=(x_i;{\gamma }_i,{\delta }_i),i=1,2.$ Applying (43), the system to be solved is

$$\left.\begin{array}{ccc}\hfill -10x_1+15x_2& =\hfill & 70\hfill \\ \hfill 11x_1+20x_2& =\hfill & 69\hfill \\ \hfill 10{\gamma }_1+0.25|x_1|+15{\gamma }_2+0.5\left|x_2\right|& =\hfill & 11\hfill \\ \hfill 11{\gamma }_1+0.5|x_1|+20{\gamma }_2+0.25\left|x_2\right|& =\hfill & 12\hfill \\ \hfill 10{\delta }_1+\left|x_1\right|+15{\delta }_2& =\hfill & 20\hfill \\ \hfill 11{\delta }_1+0.6|x_1|+20{\delta }_2+0.25\left|x_2\right|& =\hfill & 26\hfill \\ \hfill {\gamma }_1,{\gamma }_2,{\delta }_1,{\delta }_2\ge 0.& \hfill & \end{array}\right\}$$

(47)

The solution is ${\tilde{x}}_1=(-1;0.5,0.4),{\tilde{x}}_2=(4;0.25,1).$

In general, the feasibility of the FES is not guaranteed, so we can resort to an approximate system of equations:

$$\left(\mathrm{AFES}\right)\underset{j=1}{\overset{n}{⨁}}{\tilde{a}}_{ij}\otimes {\tilde{x}}_j={\tilde{b}}_i+{\widetilde{ϵ}}_i,1\le i\le n,$$

(48)

where ${\widetilde{ϵ}}_i=(0;{ϵ}_i^L,{ϵ}_i^R),{ϵ}_i^L,{ϵ}_i^R\ge 0.$

In this system, we modify the terms $b_i$ of (43). The objective is to solve (48) by minimizing the sum of modifications with respect to (43), so we use the following linear model:

$$\begin{array}{cccc}\hfill & min\hfill & \sum _{i=1}^n({ϵ}_i^L+{ϵ}_i^R)\hfill & \\ \hfill & \mathrm{s}.\mathrm{t}.\hfill & \sum _{j=1}^n{a}_{ij}{x}_j=b_i,\hfill & 1\le i\le n\hfill \\ \hfill & \hfill & \sum _{j=1}^n\left(|a_{ij}|{\gamma }_j+\left|x_j\right|{\alpha }_{ij}\right)=p_i+{ϵ}_i^L,\hfill & 1\le i\le n\hfill \\ \hfill & \hfill & \sum _{j=1}^n\left(|a_{ij}|{\delta }_j+\left|x_j\right|{\beta }_{ij}\right)=q_i+{ϵ}_i^R,\hfill & 1\le i\le n\hfill \\ \hfill & \hfill & a_j,b_j,{\alpha }_j,{\beta }_j,{\gamma }_j,{\delta }_j\ge 0,\hfill & 1\le j\le n\hfill \end{array}$$

(49)

From the solution of (49), $x_j^{\ast },{\gamma }_j^{\ast },{\delta }_j^{\ast },$$1\le j\le n,$ we construct the triangular fuzzy numbers

$${\tilde{x}}_j^{\ast }=(x_j^{\ast };{\alpha }_j^{\ast },{\beta }_j^{\ast }),1\le i\le n.$$

(50)

which are a solution to (48).

Example 4.

Solve the following system of equations:

$$\left.\begin{array}{ccc}(10;0.5,0.5)\otimes \widetilde{x_1}+(15;0.375,0.5)\otimes \widetilde{x_2}\hfill & =& (40;9,8)\hfill \\ (11;0.5,0.3)\otimes \widetilde{x_1}+(20;1,0.25)\otimes \widetilde{x_2}\hfill & =& (51;13,9)\hfill \end{array}\right\}$$

(51)

Considering (43), the solution to the auxiliary system is

$$\begin{array}{ccc}{x}_1=1,\hfill & {\gamma }_1=-0.0714,\hfill & {\delta }_1=0.2000,\hfill \\ x_2=2,\hfill & {\gamma }_2=0.5643,\hfill & {\delta }_2=0.3000.\hfill \end{array}$$

(52)

Since ${\gamma }_1<0$, the system (51) has no solution, so we resort to an approximate solution (AFES), which is solved according to (49).

$$\begin{array}{cccc}\mathrm{Min}\hfill & z={ϵ}_1^L+{ϵ}_1^R+{ϵ}_2^L+{ϵ}_2^R\hfill & \hfill & \\ \mathrm{s}.\mathrm{a}.\hfill & 10x_1+15x_2\hfill & =\hfill & 40\hfill \\ \hfill & 11x_1+20x_2\hfill & =\hfill & 51\hfill \\ \hfill & 10{\gamma }_1+0.5x_1+15{\gamma }_2+0.375x_2\hfill & =\hfill & 9+{ϵ}_1^L\hfill \\ \hfill & 11{\gamma }_1+0.5x_1+20{\gamma }_2+x_2\hfill & =\hfill & 13+{ϵ}_2^L\hfill \\ \hfill & 10{\delta }_1+0.5x_1+15{\delta }_2+0.5x_2\hfill & =\hfill & 8+{ϵ}_1^R\hfill \\ \hfill & 11{\delta }_1+0.3x_1+20{\delta }_2+0.25x_2\hfill & =\hfill & 9++{ϵ}_2^R\hfill \\ \hfill & {ϵ}_1^L,{ϵ}_2^L,{ϵ}_1^R,{ϵ}_2^R,{\gamma }_1,{\gamma }_2,{\delta }_1,{\delta }_2\hfill & \ge \hfill & 0\hfill \end{array}$$

(53)

The solution is

$$\begin{array}{cccccc}{z}^{\ast }=0.125\hfill & x_1^{\ast }=1,\hfill & {\gamma }_1^{\ast }=0.000,\hfill & {\delta }_1^{\ast }=0.200,\hfill & {ϵ}_1^{L\ast }=0.125,\hfill & {ϵ}_1^{R\ast }=0.000,\hfill \\ \hfill & x_2^{\ast }=2,\hfill & {\gamma }_2^{\ast }=0.525,\hfill & {\delta }_2^{\ast }=0.300,\hfill & {ϵ}_2^{L\ast }=0.000,\hfill & {ϵ}_2^{R\ast }=0.000.\hfill \end{array}$$

(54)

Therefore, ${\tilde{x}}_1=(1;0,0.2),{\tilde{x}}_2=(2;0.525,0.3),$ is the solution to the system

$$\left.\begin{array}{ccc}(10;0.5,0.5)\otimes \widetilde{x_1}+(15;0.375,0.5)\otimes \widetilde{x_2}\hfill & =& (40;9.125,8)\hfill \\ (11;0.5,0.3)\otimes \widetilde{x_1}+(20;1,0.25)\otimes \widetilde{x_2}\hfill & =& (51;13,9)\hfill \end{array}\right\}.$$

(55)

Remark 5.

In this work, the linear programming problems have been using LINGO (https://www.lingo.com/), the mathematical modeling language developed by LINDO Systems Inc., Chicago, IL, USA, to solve optimization problems, including linear, nonlinear, and integer problems.

4.4. Comparison with Other Methods

Before addressing applications to real-world cases, we consider it relevant to compare our proposal with other approaches that deal with the solution of equilibrium equations from the perspective of fuzzy logic. The following table compares our proposal with other approaches found in the literature. In order, the columns of

Table 2. Comparative summary of selected fuzzy and uncertain linear systems.

Reference	Type of Fuzzy System	Methodology	Dynamic?	Magnitude?	Applications
Our proposal	Fuzzy coefficients, right-hand sides, and/or variables (LR numbers)	Fuzzy linear systems, $\alpha$-cuts resolution, matrix analysis, and real-world case studies	No	Yes	Market equilibrium, applied economics, and engineering
Ahmad et al. (2025) [11]	Stochastic (Brownian noise and bifurcations)	Stochastic differential equations, Monte Carlo simulations, Hebbian dynamics, and solution analysis	Yes	No	Financial modeling and option pricing
Singh et al. (2024) [6]	Intuitionistic triangular fuzzy systems	Intuitionistic fuzzy extension, $\alpha$-cut-based decomposition, and precision analysis	No	Partially	Multi-criteria decision-making
Ullah et al. (2023) [12]	Fractional and fuzzy differential equations	Fuzzy Laplace transform, Adomian decomposition, and varying fractional orders	Yes	Partially	Mathematical biology and fuzzy epidemiology
Abbasi & Allahviranloo (2022) [8]	Fully fuzzy system using transmission average (TA)	TA-based solution approach, avoiding Zadeh’s extension principle, and uniqueness/ existence proofs	No	Partially	Engineering and uncertainty optimization
Mikaeilvand et al. (2020) [10]	Fuzzy coefficients (embedding method)	System embedding into two crisp systems and computational performance comparison	No	No	Physics, engineering, and economics
Inearat & Qatanani (2018) [9]	Triangular fuzzy right-hand side	Iterative methods: Jacobi, Gauss-Seidel, and SOR; empirical convergence analysis	No	No	Fuzzy numerical analysis
Pandit (2013) [26,27]	Fully fuzzy with negative fuzzy parameters	$\alpha$-cuts decomposition and parallel crisp system resolution for negative fuzzy terms	No	No	Fuzzy economics
Ezzati (2011) [16]	Fully fuzzy (classic embedding method)	Transformation into two crisp systems and simplified embedding method	No	No	Engineering and statistics
Friedman et al. (1998) [21]	Variables and right-hand side fuzzy	Transformation into an auxiliary crisp system	No	No	Engineering

include the type of fuzzy system, the methodology used, whether dynamic systems are addressed, whether the magnitude of uncertainty is considered, and the applications.

As shown in Table 2, the advantages of our method are twofold. First, it allows for the possibility that any component of the system (coefficients, variables, or the right-hand side) may be represented as a fuzzy number. Second, when the system is subject to a low degree of uncertainty (i.e., small tolerances) or is infeasible, a mathematical programming model is proposed to obtain a feasible solution. Nevertheless, a limitation of our approach is that it does not consider dynamic models.

In the following sections, we will present applications of our proposal to the equilibrium of supply and demand, but our approach could be applied to other situations in economics and finance.

(a)

Input–output models of the Leontief type [1,2], in which the data exhibit uncertainty and can be expressed through fuzzy systems of equations. Specifically,

–

Computation of the total production required in each economic sector to satisfy a given final demand, taking into account the interdependencies among sectors when data or equations are expressed imprecisely.

–

Renewable energy production models, where the data are inherently affected by uncertainty.

–

Extension to dynamic production models involving interrelated sectors.

(b)

Income models [2], in which some parameters may be expressed using fuzzy numbers.

(c)

Computation of dynamic equilibrium prices [3], where the use of fuzzy formalism allows for broader applicability.

5. Application to the Equilibrium of Supply and Demand

In 1874, Léon Walras (1834–1910) developed the concept of competitive general equilibrium in his work Éléments d’économie politique pure, ou théorie de la richesse sociale. The starting point is an economy with n goods and n markets, where prices are determined solely by the interaction between supply and demand (perfect competition), meaning that firms lack the power to manipulate prices. The goal is to achieve simultaneous equilibrium across all markets.

Definition 6.

Consider an economy with n goods whose prices are given by the price vector $p=(p_1,p_2,\dots ,p_n)$. For each good, $i=1,\dots ,n$, let $D_i\left(p\right)$ denote the aggregate demand for good i, $S_i\left(p\right)$ the aggregate supply for good i, and $E_i\left(p\right)=D_i\left(p\right)-S_i\left(p\right)$ the excess demand. A price vector $p^{\ast }$ is in Walrasian general equilibrium if

$$E_i\left(p^{\ast }\right)=01\le i\le n.$$

(56)

A practical way to reach equilibrium is to establish a price adjustment process, known as tâtonnement [3], which can be expressed algorithmically as follows:

STEP 1:

Propose a provisional price vector.

STEP 2:

Agents respond with their offered and demanded quantities.

STEP 3:

If there is excess demand, the price increases; if there is excess supply, the price decreases.

STEP 4:

The process repeats until all markets reach equilibrium simultaneously.

In this work, we address the supply–demand general equilibrium (56) in cases where it can be expressed through a system of linear equations.

$$\sum _{j=1}^n{a}_{ij}{p}_j=b_i,1\le i\le n,p_j\ge 0,1\le j\le n.$$

(57)

In addition to the computational simplicity of Model (57), all its elements are easy to interpret:

-

The coefficient $a_{ij}$ indicates the extent to which the price of good j influences market i to achieve equilibrium prices.

-

The unknown $p_j$ represents the price of good j required for equilibrium across the n markets.

-

The independent term, $b_i$, represents the fixed excess demand (when supply and demand do not depend on prices) in the i-th market.

Normally, Equation (57) is established through statistical estimations [1,2,3]. This means that the value of the coefficients may be affected by imprecision and/or uncertainty, which can influence the solution of the model [2]. Moreover, the prices themselves (the unknowns of the system) may also be considered as vague quantities [17,26,27]. This makes the formulation using fuzzy numbers a very suitable alternative to reflect the supply–demand equilibrium in the real world.

In a supply–demand equilibrium model expressed through fuzzy systems, the fuzzy equilibrium represents a fuzzy set of prices and quantities in which an approximate equilibrium is achieved, taking into account the uncertainty or imprecision of the model. That is, an equilibrium zone is obtained [26,27] instead of a single point, as occurs in th classical equilibrium [1]. This reflects that economic agents (producers and consumers) do not have perfect information, or that their preferences, costs, or incomes are imprecisely defined.

Next, in

Table 3. Types of fuzzy market equilibrium models.

	Formulation	Uncertainty/Imprecision
Scenario 1	${⨁}_{j=1}^n{\tilde{a}}_{ij}{p}_j={\tilde{b}}_i,$$1\le i\le n.$	- Influence of prices on market equilibrium; - Fixed excess demand.
Scenario 2	${⨁}_{j=1}^n{a}_{ij}{\tilde{p}}_j={\tilde{b}}_i,$$1\le i\le n.$	- Prices; - Fixed excess demand.
Scenario 3	${⨁}_{j=1}^n{\tilde{a}}_{ij}\otimes {\tilde{p}}_j={\tilde{b}}_i,$$1\le i\le n.$	- Influence of prices on market equilibrium; - Prices; - Fixed excess demand.

, we present all the possible formulations of the supply–demand equilibrium using fuzzy systems of equations.

To illustrate all the cases, we present below two applications that are a simplification of real cases. The first one is based on a Colombian multinational coffee company, and the second on a Spanish paper company. In these applications, we can analyze all the cases presented in Table 3.

5.1. Case I: Colombian Multinational Coffee Company

The demand for coffee depends on its price, the price of tea (a substitute for coffee), and the price of milk (a complement to coffee). A specialty firm operating in three neighborhoods, A, B, and C, with different socioeconomic conditions, aims to establish neighborhood-independent prices. To do so, it focuses on the most demanded product in the establishment, coffee, and estimates that the supply and demand for coffee are given by

$$\begin{array}{cc}{D}_A\approx 50-2p_c+0.5p_t,\hfill & S_A\approx 20+3p_c-p_t-2p_l\hfill \\ D_B\approx 58-1.5p_c,\hfill & S_B\approx 15+3.5p_c-0.5p_t-1.75p_l\hfill \\ D_C\approx 45-2.5p_c+1.5p_l\hfill & S_C\approx 20+1.5p_c-p_t-1.5p_l\hfill \end{array}$$

(58)

where $p_c$ is the price of 1 kg of coffee, $p_t$ is the price of 200 g of tea, and $p_l$ is the price of 1 L of milk, all expressed in dollars.

The numerical data are real and were provided by the company, although the problem has been simplified by removing products and neighborhoods to make the graphs and systems easier to understand. On the other hand, the uncertainty present in each of the scenarios was determined after consulting three experts employed by the company.

Scenario 1.

Fuzzy independent terms and unknowns: According to [21], we assume that the uncertainty arises from the unknowns (in this case, the prices) and the fixed terms, but not from the coefficients of the unknowns.

$$\begin{array}{cc}\widetilde{D_A}=\widetilde{50}-2{\tilde{p}}_c+0.5{\tilde{p}}_t,\hfill & \widetilde{S_A}=\widetilde{20}+3{\tilde{p}}_c-1{\tilde{p}}_t-2{\tilde{p}}_l\hfill \\ \widetilde{D_B}=\widetilde{58}-1.5{\tilde{p}}_c,\hfill & \widetilde{S_B}=\widetilde{15}+3.5{\tilde{p}}_c-0.5{\tilde{p}}_t-1.75{\tilde{p}}_l\hfill \\ \widetilde{D_C}=\widetilde{45}-2.5{\tilde{p}}_c+1.5{\tilde{p}}_l\hfill & \widetilde{S_C}=\widetilde{20}+1.5{\tilde{p}}_c-{\tilde{p}}_t-1.5{\tilde{p}}_l\hfill \end{array}$$

(59)

If we model the uncertainty using triangular fuzzy numbers $\widetilde{50}=(45,50,52),$$\widetilde{58}=(55,58,63),\widetilde{45}=(40,45,50),\widetilde{20}=(18,20,21),\widetilde{15}=(13,15,19),$ and set ${\tilde{S}}_A={\tilde{D}}_A$, ${\tilde{S}}_B={\tilde{D}}_B$ and ${\tilde{S}}_C={\tilde{D}}_C$, according to [21], we propose solving the following model:

$$\left.\begin{array}{ccc}5{\tilde{p}}_c-1.5{\tilde{p}}_t-2{\tilde{p}}_l\hfill & =\hfill & [24+6\alpha ,34-4\alpha ]\hfill \\ 5{\tilde{p}}_c-0.5{\tilde{p}}_t-1.75{\tilde{p}}_l\hfill & =\hfill & [36+7\alpha ,50-7\alpha ]\hfill \\ 4{\tilde{p}}_c-{\tilde{p}}_t-3{\tilde{p}}_l\hfill & =\hfill & [19+6\alpha ,32-7\alpha ]\hfill \end{array}\right\}$$

(60)

which is solved using the system of Linear Equation (61).

$$\left(\begin{array}{cccccc}5& 0& 0& 0& 1.5& 2\\ 5& 0& 0& 0& 0.5& 1.75\\ 4& 0& 0& 0& 1& 3\\ 0& 1.5& 2& 5& 0& 0\\ 0& 0.5& 1.75& 5& 0& 0\\ 0& 1& 3& 4& 0& 0\end{array}\right)\left(\begin{array}{c}{p}_c^L\\ p_t^L\\ p_l^L\\ -p_c^R\\ -p_t^R\\ -p_l^R\end{array}\right)=\left(\begin{array}{c}24+6\alpha \\ 36+7\alpha \\ 19+6\alpha \\ -34+4\alpha \\ -50+7\alpha \\ -32+7\alpha \end{array}\right)$$

(61)

Considering (28), the solution of (61), expressed in terms of $\alpha$-cuts, is

$$\begin{array}{c}{\tilde{p}}_c\left(\alpha \right)=[8.983+1.276\alpha ,11.242-0.983\alpha ],\hfill \\ {\tilde{p}}_t\left(\alpha \right)=[11.552+1.172\alpha ,16.276-3.552\alpha ],\hfill \\ {\tilde{p}}_l\left(\alpha \right)=[-1.103+2.207\alpha ,1.793-0.690\alpha ]\hfill \end{array}$$

(62)

As can be seen in Figure 3, the solution only makes sense as equilibrium prices for $\alpha \in ]0.499,1]$, since the price $p_l$ would be negative for values of $\alpha$ less than 0.499.

Scenario 2.

Coefficients with uncertainty: According to the proposal by Carlsson and Korhonen in [20], we assume that the uncertainty lies in some coefficients of the supply and demand functions, where the use of braces follows the notation expressed in Section 4.1; i.e., $\{a,b\}$ means that a is more plausible than b.

$$\begin{array}{cc}{\tilde{D}}_A=\{45,55\}-2p_c+\{1,0.75\}{p}_t,\hfill & {\tilde{S}}_A=\{21,18\}+\{3,3.5\}{p}_c-p_t-2p_l\hfill \\ {\tilde{D}}_B=\{58,60\}-1.5p_c,\hfill & {\tilde{S}}_B=\{16,14\}+\{3.75,3.5\}{p}_c-0.5p_t-1.75p_l\hfill \\ {\tilde{D}}_C=\{43,46\}-2.5p_c+\{1.75,2.25\}{p}_l\hfill & {\tilde{S}}_C=\{21,19\}+\{2,1.25\}{p}_c-p_t-1.5p_l.\hfill \end{array}$$

(63)

Following the approach in Section 4.1, the auxiliary system of equations to be solved is Parametric System (64), where $\alpha \in [0,1]$.

$$\left.\begin{array}{c}(5.5-0.5\alpha )p_c+(-1.75-0.25\alpha )p_t-2p_l=37-13\alpha ,\hfill \\ (5+0.25\alpha )p_c-0.5p_t-1.75p_l=46-4\alpha \hfill \\ (3.75+0.75\alpha )p_c-p_t+(-3.75+0.5\alpha )p_l=26-5\alpha \hfill \end{array}\right\}$$

(64)

The solutions, as a function of the parameter $\alpha$, are

$$\begin{array}{cc}{p}_c\left(\alpha \right)={\displaystyle \frac{-32{\alpha }^3+452{\alpha }^2+356\alpha -9704}{2{\alpha }^3+68{\alpha }^2+19\alpha -921}},\hfill & p_t\left(\alpha \right)={\displaystyle \frac{168{\alpha }^3+232{\alpha }^2+2276\alpha -10404}{2{\alpha }^3+68{\alpha }^2+19\alpha -921}},\hfill \\ p_l\left(\alpha \right)={\displaystyle \frac{28{\alpha }^3-468{\alpha }^2-3624\alpha -544}{2{\alpha }^3+68{\alpha }^2+19\alpha -921}}.\hfill & \end{array}$$

(65)

For instance, a price consistent with the traditional market could be $p_c\left(0.15\right)=10.5$$, $p_t\left(0.15\right)=11.1$$, $p_l\left(0.15\right)=1$$ (Figure 4).

Scenario 3.

Fuzzy coefficients, independent terms, and unknowns: We will assume that there is uncertainty in all the estimates that appear in the supply and demand functions.

$$\begin{array}{cc}\widetilde{D_A}=\widetilde{50}-\tilde{2}{\tilde{p}}_c+\widetilde{0.5}{\tilde{p}}_t,\hfill & \widetilde{S_A}=\widetilde{20}+\tilde{3}{\tilde{p}}_c-\tilde{1}{\tilde{p}}_t-\tilde{2}{\tilde{p}}_l,\hfill \\ \widetilde{D_B}=\widetilde{58}-\widetilde{1.5}{\tilde{p}}_c,\hfill & \widetilde{S_B}=\widetilde{15}+\widetilde{3.5}{\tilde{p}}_c-\widetilde{0.5}{\tilde{p}}_t-\widetilde{1.75}{\tilde{p}}_l,\hfill \\ \widetilde{D_C}=\widetilde{45}-\widetilde{2.5}{\tilde{p}}_c+\widetilde{1.5}{\tilde{p}}_l\hfill & \widetilde{S_C}=\widetilde{20}+\widetilde{1.5}{\tilde{p}}_c-\tilde{1}{\tilde{p}}_t-\widetilde{1.5}{\tilde{p}}_l,\hfill \end{array}$$

(66)

$$\begin{array}{c}\widetilde{D_A}=(45,50,52)\oplus (-2.5,-2,-1.75)\otimes {\tilde{p}}_c\oplus (0.5,0.5,0.75)\otimes {\tilde{p}}_t,\hfill \\ \widetilde{D_B}=(55,58,63)\oplus (-1.75,-1.5,-1)\otimes {\tilde{p}}_c,\hfill \\ \widetilde{D_C}=(40,45,50)\oplus (-2.75,-2.5,-2)\otimes {\tilde{p}}_c\oplus (1,1.5,1.75)\otimes {\tilde{p}}_l\hfill \\ \widetilde{S_A}=(18,20,21)\oplus (2.5,3,3.5)\otimes {\tilde{p}}_c\oplus (-1.5,-1,-0.75)\otimes {\tilde{p}}_t,\oplus (-2.5,-2,-1.5)\otimes {\tilde{p}}_l\hfill \\ \widetilde{S_B}=(13,15,19)\oplus (3,3.5,3.75)\otimes {\tilde{p}}_c\oplus (-1,-0.5,-0.25)\otimes {\tilde{p}}_t,\oplus (-2,-1.75,-1.25)\otimes {\tilde{p}}_l,\hfill \\ \widetilde{S_C}=(18,20,21)\oplus (1,1.5,2)\otimes {\tilde{p}}_c\oplus (-1.5,-1,-0.75)\otimes {\tilde{p}}_t\oplus (-2,-1.5,-1.25)\otimes {\tilde{p}}_l,\hfill \end{array}$$

(67)

Setting ${\tilde{S}}_A={\tilde{D}}_A$, ${\tilde{S}}_B={\tilde{D}}_B$ and ${\tilde{S}}_C={\tilde{D}}_C$ and expressing the fuzzy numbers as $\alpha$-cuts, the fuzzy system to be solved is

$$\left.\begin{array}{c}(4.25+0.75\alpha ,5.25-0.25\alpha )(p_c^L,p_c^R)+(-2.25+0.75\alpha ,-1.25-0.25\alpha )(p_t^L,p_t^R)\hfill \\ +(-2.5+0.5\alpha ,-1.5-0.5\alpha )(p_l^L,p_l^R)=(24+6\alpha ,34-4\alpha )\hfill \\ (4+\alpha ,5.5-0.5\alpha )(p_c^L,p_c^R)+(-1+0.5\alpha ,-0.25-0.25\alpha )(p_t^L,p_t^R)\hfill \\ +(-2+0.25\alpha ,-1.25-0.5\alpha )(p_l^L,p_l^R)=(36+7\alpha ,50-7\alpha )\hfill \\ (3+\alpha ,4.75-0.75\alpha )(p_c^L,p_c^R)+(-1.5+0.5\alpha ,-0.75-0.25\alpha )(p_t^L,p_t^R)\hfill \\ +(-3.75+0.75\alpha ,-2.25-0.75\alpha )(p_l^L,p_l^R)=(19+6\alpha ,32-7\alpha )\hfill \end{array}\right\}$$

(68)

According to (33), to find the solution to System (68), we solve the following auxiliary system:

$$\left(\begin{array}{cc}{A}_{3\times 3}\left(\alpha \right)& 0\\ 0& B_{3\times 3}\left(\alpha \right)\end{array}\right)\left(\begin{array}{c}{p}_c^L\\ p_t^L\\ p_l^L\\ p_c^R\\ p_t^R\\ p_l^R\end{array}\right)=\left(\begin{array}{c}24+6\alpha \\ 36+7\alpha \\ 19+6\alpha \\ 34-4\alpha \\ 50-7\alpha \\ 32-7\alpha \end{array}\right)$$

(69)

where the matrices $A_{3\times 3}\left(\alpha \right)$ and $B_{3\times 3}\left(\alpha \right)$ are

$$A_{3\times 3}\left(\alpha \right)=\left(\begin{array}{ccc}4.25+0.75\alpha & -1.25-0.25\alpha & -1.5-0.5\alpha \\ 4+\alpha & -0.25-0.25\alpha & -1.25-0.5\alpha \\ 3+\alpha & -0.75-0.25\alpha & -2.25-0.75\alpha \end{array}\right)$$

(70)

$$B_{3\times 3}\left(\alpha \right)=\left(\begin{array}{ccc}5.25-0.25\alpha & -2.25+0.75\alpha & -2.5+0.5\alpha \\ 5.5-0.5\alpha & -1+0.5\alpha & -2+0.25\alpha \\ 4.75-0.75\alpha & -1.5+0.5\alpha & -3.75+0.75\alpha \end{array}\right)$$

(71)

Note that in matrices $A_{3\times 3}\left(\alpha \right)$ and $B_{3\times 3}\left(\alpha \right)$, according to Step 4 of Section 4.2.2, the terms $a_{ij}$ have been swapped with $b_{ij}$ for $j=2,3$.

$$\begin{array}{c}{\displaystyle p_c\left(\alpha \right)=\left[\frac{4{\alpha }^3+192{\alpha }^2+1544\alpha +3020}{{\alpha }^3+16{\alpha }^2+141\alpha +306},\frac{104{\alpha }^3-1244{\alpha }^2+5652\alpha -9272}{9{\alpha }^3-90{\alpha }^2+407\alpha -790}\right]}\hfill \\ {\displaystyle p_t\left(\alpha \right)=\left[\frac{-48{\alpha }^3+212{\alpha }^2+1380\alpha -7448}{9{\alpha }^3-90{\alpha }^2+407\alpha -790},\frac{-20{\alpha }^3-40{\alpha }^2+1528\alpha +4436}{{\alpha }^3+16{\alpha }^2+141\alpha +306}\right]}\hfill \\ {\displaystyle p_l\left(\alpha \right)=\left[\frac{4{\alpha }^3+140{\alpha }^2+404\alpha -36}{{\alpha }^3+16{\alpha }^2+141\alpha +306},\frac{52{\alpha }^3-656{\alpha }^2+2116\alpha -2024}{9{\alpha }^3-90{\alpha }^2+407\alpha -790}\right]}\hfill \end{array}$$

(72)

In Figure 5 the membership functions of the solutions ${\tilde{p}}_c,{\tilde{p}}_t,{\tilde{p}}_l$ obtained from the system of equations (in black) and the adjusted membership functions for triangular fuzzy numbers (in orange) are depicted.

$${\tilde{p}}_c=(9.809,1.103,10.259),{\tilde{p}}_t=(9.428,12.724,14.497),{\tilde{p}}_l=(0,1.104,2.562).$$

(73)

Similar to what happened in Scenario 1, for $p_l$ to be positive, $\alpha$ must be in the interval [0.086, 1].

5.2. Case II: Spanish Paper Company

A paper company with a strong presence in the Valencian community (Spain) sells three products in supermarkets: labels, A; planners, B; and notepads, C. An analysis by the company shows that its supply and demand capacities in three stores for nit prices $p_1,p_2$, and $p_3$ in dollars (respectively) are given (in thousands of units) by

$$\begin{array}{ccc}\mathrm{Store}1:\hfill & S_A=10+2.5p_1+p_2+p_3,\hfill & D_A=39-3p_1-2p_2,\hfill \\ \mathrm{Store}2:\hfill & S_B=12+p_1+2.5p_2+p_3,\hfill & D_B=62-2p_2-p_3,\hfill \\ \mathrm{Store}3:\hfill & S_C=7+p_1+p_2+2.5p_3,\hfill & D_C=47-p_1-4p_3.\hfill \end{array}$$

(74)

The numerical data are estimates provided by the company, which was not aware of the model’s infeasibility. The problem has been simplified by removing products and stores to facilitate understanding. On the other hand, when the two company managers became aware of the infeasibility of their estimation, they identified which elements and how much uncertainty uncertainty could be introduced into the model.

The goal is to determine common prices for the three stores, considering supply and demand. Equating supply to demand, we obtain the following system:

$$\left.\begin{array}{ccc}5.5p_1+3p_2+p_3& =& 29\\ p_1+4.5p_2+2p_3& =& 50\\ 2p_1+p_2+6.5p_3& =& 40\end{array}\right\}$$

(75)

Since the solution is $p_1=-{\displaystyle \frac{598}{1075}},p_2={\displaystyle \frac{1944}{215}},$ and $p_3={\displaystyle \frac{5304}{1075}},$ it is evident that equilibrium prices do not exist due to the negative price. Below, we address this situation through two scenarios.

Scenario 4.

Coefficients with uncertainty: To resolve this issue, a new market study (which is costly and time-consuming) could be conducted, or additional information could be extracted from the company’s managers. First, the managers know that they could attempt modifications in supply (which depends on the company), but not in demand. Observing the functions $S_A,S_B$ and $S_C$, it is clear that the supply of each product depends $2.5$ times more on the price of the product itself that on the other two. Given this, information is requested to formulate this proportion as an interval in the sense of (15). For A, B, and C, the proportion is estimated to be in the intervals $\{2,3\},\{3.5,2.5\},$ and $\{3,2.5\}$, respectively, where according to (15), the first value in the interval is the most plausible and the second is the least plausible. Following (19), we express the coefficients as a function of a single parameter (20), where

$$\begin{array}{c}{S}_A\left(\alpha \right)=10+(3-\alpha )p_1+p_2+p_3,\hfill \\ S_B\left(\alpha \right)=12+p_1+(2.5+\alpha )p_2+p_3,\hfill \\ S_C\left(\alpha \right)=7+p_1+p_2+(2.5+\alpha )p_3.\hfill \end{array}$$

(76)

With these supply functions, the system becomes

$$\left.\begin{array}{ccc}(6-\alpha )p_1+3p_2+p_3& =& 29\\ p_1+(4.5+\alpha )p_2+2p_3& =& 50\\ 2p_1+p_2+(6.5+\alpha )p_3& =& 40\end{array}\right\}$$

(77)

The solution, as a function of the parameter $\alpha$, is

$$\begin{array}{cc}{\displaystyle p_1\left(\alpha \right)=\frac{-116{\alpha }^2-516\alpha +299}{4{\alpha }^3+20{\alpha }^2-135\alpha -592},}\hfill & {\displaystyle p_2\left(\alpha \right)=\frac{200{\alpha }^2-104\alpha -5350}{4{\alpha }^3+20{\alpha }^2-135\alpha -592},}\hfill \\ {\displaystyle p_3\left(\alpha \right)=\frac{160{\alpha }^2-208\alpha -2912}{4{\alpha }^3+20{\alpha }^2-135\alpha -592},}\hfill & \alpha \in [0,1].\hfill \end{array}$$

(78)

Consequently, equilibrium prices can be established for values of $\alpha \in \left]{\displaystyle \frac{-129+2\sqrt{1582}}{58}},1\right]\approx ]0.5189,1],$ as all prices are positive. For example, for $\alpha =0.65$, the equilibrium prices are

$$p_1\left(0.625\right)\approx 0.1\$,p_2\left(0.625\right)\approx 8\$,p_3\left(0.625\right)\approx 4.5\$.$$

(79)

The supply functions in (80) are quite similar to the original ones in (74), where

$$\begin{array}{cc}\hfill S_A\left(0.625\right)& =10+2.375p_1+p_2+p_3,\hfill \\ \hfill S_B\left(0.625\right)& =12+p_1+3.125p_2+p_3,\hfill \\ \hfill S_C\left(0.625\right)& =7+p_1+p_2+3.125p_3.\hfill \end{array}$$

(80)

Scenario 5.

Fuzzy systems with small tolerances: As in Scenario 4, we will make modifications to the supply functions, but this time, according to Section 4.3, we will consider small tolerances:

$$\begin{array}{c}{S}_A=10+(2.5;0.1,0.2)p_1+p_2+p_3,\hfill \\ S_B=12+p_1+(2.5;0.1,0.1)p_2+p_3,\hfill \\ S_C=7+p_1+p_2+(2.5;0.1,0.1)p_3.\hfill \end{array}$$

(81)

With this, the system to be solved is

$$\left.\begin{array}{ccc}(6;0.1,0.2)\otimes {\tilde{p}}_1+3{\tilde{p}}_2+{\tilde{p}}_3& =& (29;0,0)\\ {\tilde{p}}_1+(5;0.1,0.1)\otimes {\tilde{p}}_2+2{\tilde{p}}_3& =& (50;0,0)\\ 2{\tilde{p}}_1+{\tilde{p}}_2+(6.5;0.1,0.1)\otimes {\tilde{p}}_3& =& (40;0,0)\end{array}\right\}$$

(82)

Obviously, (82) cannot provide a fuzzy solution, since the tolerances of the independent terms are zero. The solution to (82) is

$$\begin{array}{c}{p}_1=-0.0030,p_2=8.0330,p_3=4.9189\hfill \\ {\gamma }_1=-0.00005,{\delta }_1=-0.0001,{\gamma }_2={\delta }_2=-0.16767,{\gamma }_3={\delta }_3=-0.07568.\hfill \end{array}$$

(83)

Taking into account (49), we can calculate the minimum sum of tolerances that provide a fuzzy solution to a system approximated to (82), as indicated in (48).

$$\mathrm{Min}\left\{\begin{array}{cc}\sum _{i=1}^3{ϵ}_i^L:\hfill & 6p_1+3p_2+p_3=29,p_1+5p_2+2p_3=50,2p_1+p_2+6.5p_3=40,\hfill \\ \hfill & 6{\gamma }_1+0.1|p_1|={ϵ}_1^L,5{\gamma }_2+0.1|p_2|={ϵ}_2^L,6.5{\gamma }_3+0.1\left|p_3\right|={ϵ}_3^L,\hfill \\ \hfill & 6{\delta }_1+0.2|p_1|={ϵ}_1^R,5{\delta }_2+0.1|p_2|={ϵ}_2^R,6.5{\delta }_3+0.1\left|p_3\right|={ϵ}_3^R,\hfill \\ \hfill & {\gamma }_i,{\delta }_i\ge 0,i=1,2,3.\hfill \end{array}\right\}$$

(84)

The fuzzy solution provided by (84) is

$${\tilde{p}}_1=(-0.0030;0,0.2056),{\tilde{p}}_2=(8.3003;0,0.0862),{\tilde{p}}_3=(4.9189;0,0.1143),$$

(85)

and is obtained with the supply functions

$$\begin{array}{c}{S}_A=(10;1.2346,0.003)+(2.5;0.1,0.2)p_1+p_2+p_3,\hfill \\ S_B=(12;1.2346,0)+p_1+(2.5;0.1,0.1)p_2+p_3,\hfill \\ S_C=(7;1.2346,0)+p_1+p_2+(2.5;0.1,0.1)p_3.\hfill \end{array}$$

(86)

Note that the solution provided in (85) and in (78) are very similar and that the demand functions are quite similar to the original ones given in (74).

6. Results

The results obtained demonstrate the ability of fuzzy linear equation systems to model and solve real economic situations characterized by a high degree of uncertainty. Specifically, the usefulness of different model formulations (with uncertainty in coefficients, independent terms, and/or variables) has been validated using $LR$-fuzzy numbers.

Specific algorithms were developed and applied by converting the fuzzy system into a crisp parametric system using $\alpha$-cuts. This methodology enabled both numerical and graphical interpretation of the solutions and allowed the use of classical matrix algebra techniques. To implement this, it was necessary to reformulate the concept of a weak solution.

One of the main contributions of this work lies in the practical application of the model to two real (simplified) multiproduct market equilibrium problems under uncertainty.

In the first case, the price equilibrium of coffee, tea, and milk was analyzed within a Colombian multinational company. Demand and supply functions were formulated by neighborhood (A, B, and C), incorporating the cross-price effects between complementary and substitute products. The uncertainty in the estimates was modeled using triangular fuzzy numbers, allowing the variability inherent in consumer behavior to be captured. The resulting fuzzy systems were solved using $\alpha$-cuts and linear programming, yielding fuzzy price vectors that represent plausible price ranges to achieve market equilibrium for the three products and areas analyzed.

The second case addressed the price equilibrium problem in a company within the paper industry, which operated with incomplete information derived from a market study. Due to the lack of precise data, conventional deterministic methods were not viable. The system was thus modeled as a set of fuzzy linear equations with small tolerances, and the AFES (approximate fuzzy equation system) method was applied. This approach allowed for the derivation of a fuzzy solution by minimizing the necessary adjustments to the constant terms, formulating the problem as a linear programming model. The results show that even when the original system does not admit an exact solution, it is possible to find optimal approximate solutions with minimal distortion of the initial data. This validates the model as an effective tool in contexts where information quality is low or contradictory.

From a methodological perspective, the results were compared with existing approaches in the literature. The proposed method offers significant advantages, as it generalizes previous techniques by allowing uncertainty in multiple system components and enhances the analytical potential by integrating graphical, algebraic, and mathematical programming tools for interpreting economic outcomes.

Finally, it is worth noting that while this study focuses on the food and paper industries, the model’s applicability to other sectors is immediate. For instance, in the tourism sector, hotel pricing is highly sensitive to the intermediary involved in the booking process, introducing considerable uncertainty. Similarly, in the energy sector, supply–demand equilibrium in electrical grids with renewable sources (where both generation and consumption are uncertain) requires fuzzy logic-based approaches.

7. Conclusions

A robust methodology has been presented for the analysis of economic systems under uncertainty, based on fuzzy linear equation systems. It has been demonstrated that this approach is capable of generalizing traditional deterministic models by explicitly incorporating uncertainty into their structural components.

Among the most relevant contributions are the formalization of different types of fuzzy systems (FC, FV, and FLS), the development of conditions for the existence and construction of fuzzy solutions, and the validation of the methodology through two applied cases of multiproduct market equilibrium.

The results suggest that this approach not only improves the representation of real economic behavior but also offers practical tools for analysis and forecasting in uncertain scenarios.

As future lines of research, we aim to address some of the applications described at the end of Section 4.4. We plan to incorporate additional constraints (e.g., price limits, scarce resources, or social preferences), as well as extend the approach to dynamic or nonlinear models that more fully capture the temporal evolution of markets under uncertainty.

We are currently working successfully with input–output models for solar energy production in a Spanish photovoltaic plant. We have found that fuzzy numbers are a useful tool for modeling production fluctuations. Although we are not allowed to present numerical data at this stage, in the near future we plan to develop a paper in which, using aggregated or simulated data, the usefulness of these methods can be demonstrated.