Fusion recognition of palmprint and palm vein based on modal correlation

1.   Introduction

The concept of type-2 fuzzy sets (T2 FSs) was first proposed by Zadeh ^[1] and a detailed introduction was given in ^[2]. T2 FSs are an extension of the type-1 fuzzy set (T1 FS) and further considers the fuzziness of the fuzzy set. Since the definition of T2 FSs was proposed, most scholars have mainly studied the operations and properties of T2 FSs ^[3,4]. Until the 1990s, Prof. Mendel redefined T2 FSs and proposed the type-2 fuzzy logic system (T2 FLS) ^[5,6].

As an extension of T1 FSs, T2 FSs overcome the limitations of T1 FSs in dealing with the uncertainties of actual objects. However the definition of T2 FSs is complicated, and the corresponding graph must be a spatial graph. Due to the complexity of the expression of T2 FSs, Mendel introduces the definition of interval T2 FSs (IT2 FSs) as a special case of T2 FSs. The secondary membership grade of IT2 FSs is constant one, which is more simpler than T2 FSs. In general, IT2 fuzzy logic system (IT2 FLS) are used in most theories and applications ^[7,8].

In the field of fuzzy control, solving fuzzy relation equations (FREs) plays an important role in the design of fuzzy controller and fuzzy logic reasoning. Most algorithms for solving FREs can obtain some specific solutions, such as the minimum or maximum solution ^[9], or can describe the solution theoretically ^[10]. In the existing methods, most of them are used for solving type-1 fuzzy relation equations (T1 FREs), while few methods are used for solving interval type-2 fuzzy relation equations (IT2 FREs). The main work of this paper is to propose a new method to obtain the entire solution set of IT2 FREs.

On the other hand, Prof. Cheng proposed a new matrix product-semi-tensor product (STP) of matrices, which is the generalization of the conventional matrix product and retains almost all the main properties of the conventional matrix product. As a novel mathematical technique for handling logical operations, STP has been successfully applied to logical systems ^{[11,12,13,14]} and, based on this, a new algorithm for solving FREs has been devised. For example, in T1 FREs, the STP is used to solve fuzzy relation equalities and fuzzy relation inequalities ^{[15,16,17,18]}. In type-2 fuzzy relational equations (T2 FREs), only some simple algorithms have been proposed to study the solution of type-2 single-valued fuzzy relation equations and type-2 symmetry-valued fuzzy relation equations ^[19,20]. However, the ordinary STP cannot be used directly to solve IT2 FREs. Therefore, we extend the STP to interval matrices and propose the STP of interval matrices, then discuss the solutions of IT2 FREs.

In the rest of this paper, section two introduces the basic concepts of the STP of interval matrices. Section three mainly gives the relevant definitions of interval-valued logic and gives its matrix representation. Section four discusses the solvability of IT2 FREs and designs an algorithm to solve IT2 FREs. Section five explains the viability of the proposed algorithm with a numerical example. Section six gives a brief summary of the paper.

2.   Preliminaries

2.1.   Basic concept

First, in order to express conveniently, we introduce some notations used throughout the paper.

●

where $\underline \alpha, \overline \alpha\in R$. If $\alpha = [\underline \alpha, \underline \alpha]\left({or\ {\rm { }}\alpha = [\overline \alpha, \overline \alpha]} \right)$, this is a point interval and $\alpha$ degenerates into a real number.

●

$\underline A = {\left({{{\underline \alpha }_i}} \right)_m}$ and $\overline A = {\left({{{\overline \alpha }_i}} \right)_m}$ are two m-dimensional vectors and $\left[{{{\underline a }_i}, {{\overline a }_i}} \right] \in I[0, 1]$, $i = 1 \cdots m$.

●

$\underline A = {\left({{{\underline \alpha }_{ij}}} \right)_{m \times n}}$ and $\overline A = {\left({{{\overline \alpha }_{ij}}} \right)_{m \times n}}$ are two $m \times n$ dimensional matrices and $\left[{{{\underline \alpha }_{ij}}, {{\overline \alpha }_{ij}}} \right] \in I\left[{0, 1} \right]$, $i = 1 \cdots m$, $j = 1 \cdots n$.

● $\delta _r^i$ : the ith column of unit matrix ${I_n}$.

● $\left[{\delta _r^i, \delta _r^j} \right]$ : a bounded closed interval, where $\delta _n^i$ represents its lower bound and $\delta _n^j$ represents its upper bound, abbreviated as ${\delta _r}\left[{i, j} \right]$.

● $Co{l_i}(M)$: the ith column of interval matrix $M$.

● $Ro{w_j}(M)$: the jth row of interval matrix $M$.

Next, we define $\wedge, \vee$ and $\neg$ in $I\left[{0, 1} \right]$.

Definition 2.1. ^[21] (1) Let

then,

(2) Let

then their max-min composition operation is defined as

where

where $i = 1, \cdots, m, \ j = 1, \cdots, k.$

Definition 2.2. ^[22] Let

then the partial order $\ge, \le$ and $=$ are defined as

(1) If ${\underline a _{ij}} \ge {\underline b _{ij}}, {\overline a _{ij}} \ge {\overline b _{ij}}$, we say $A \ge B.$

(2) If ${\underline a _{ij}} \le {\underline b _{ij}}, {\overline a _{ij}} \le {\overline b _{ij}}$, we say $A \le B.$

(3) If ${\underline a _{ij}}{\rm{ = }}{\underline b _{ij}}, {\overline a _{ij}}{\rm{ = }}{ \overline b _{ij}}$, we say $A{\rm{ = }}B.$

Property 2.1. ^[15] Let

Assume $A \le B$ and $C \le D$, then

Definition 2.3. ^[22] (1) Let

then the four operations of intervals $\alpha$ and $\beta$ are as follows.

1) Addition operation

2) Subtraction operation

3) Multiplication operation

4) Division operation

Note that $0 \notin \beta = [\underline \beta, \overline \beta]$.

(2) If

then the product of interval $\alpha$ and interval matrix $A$ is

(3) If

then the product of interval matrices $A$ and $B$ is

where

Based on Definition 2.3, we give the relevant definition of the STP of interval matrices.

Definition 2.4. (1) If

then the kronecker product of interval matrices $A$ and $B$ is

(2) If

then the STP of interval matrices $A$ and $B$ is

where $t = lcm\left({n, p} \right)$ is the least common multiple of $n$ and $p$.

(3) If

then the khatri-rao product of interval matrices $A$ and $B$ is

Remark 2.1. In Definition 2.4, if $n = p$, then the STP of interval matrices degenerates to the ordinary interval matrix multiplication. Therefore, the STP of interval matrices is a generalization of interval matrices multiplication. In the context, the STP of interval matrices is $\ltimes$, which is omitted by default.

Example 2.1. Given the interval matrices A and B,

The kronecker product of the interval matrix $A$ and the unit interval matrix $I_2$ is

The STP of interval matrices $A$ and $B$ is

The khatri-rao product of interval matrices $A$ and $B$ is

According to the definition of STP of interval matrices, we can get the following properties.

Property 2.2. (1) Let $A, B \in I\left({{{\left[{0, 1} \right]}^{m \times n}}} \right), C\in I\left({{{\left[{0, 1} \right]}^{p \times q}}} \right)$, then

(2) Let $A \in I\left({{{\left[{0, 1} \right]}^{m \times n}}} \right), B \in I\left({{{\left[{0, 1} \right]}^{p \times q}}} \right)$ and $C \in I\left({{{\left[{0, 1} \right]}^{r \times s}}} \right)$, then

(3) Let $A \in I\left({{{\left[{0, 1} \right]}^{m \times n}}} \right)$, $C \in I\left({{{\left[{0, 1} \right]}^s}} \right)$ and $R \in I\left({{{\left[{0, 1} \right]}^s}} \right)$ are column and row interval vectors, respectively, then

2.2.   Problem formulation

Let the interval type-2 fuzzy relation $\widetilde R \in F(V \times W)$, where the domain $V = \left\{ {{v_1}, {v_2}, \cdots, {v_n}} \right\}$ and $W = \left\{ {{w_1}, {w_2}, \cdots, {w_p}} \right\}$, then the matrix form of interval type-2 fuzzy relation $\widetilde R$ can be defined as

${\mu _{\widetilde R}}\left({{v_i}, {w_k}} \right)$ and ${f_{\widetilde R}}\left({{v_i}, {w_k}} \right)$ represent the primary membership grade and secondary membership grade of IT2 FSs, respectively. For primary membership grade, it is composed of upper membership grade and lower membership grade; that is,

The secondary membership grade of IT2 FSs equals one; that is, ${f_{\widetilde R}}\left({{v_i}, {w_k}} \right){\rm{ = }}1$, then the matrix form of interval type-2 fuzzy relation $\widetilde R$ can be further described as

Two common types of FREs exist in practical application ^[20]. One type is that the fuzzy relation is unknown, which is commonly used for designing fuzzy controllers. The other type is that the fuzzy input is unknown, which is commonly used for diagnosing diseases based on the symptom similarity. In terms of the aforementioned situations, it can be assumed that there are similar two types of IT2 FREs, as shown in Figures 1 and 2.

Type 1: assume $\widetilde A \in F(U \times V), \widetilde B \in F(U \times W)$. We seek an interval type-2 fuzzy relation $\widetilde X \in F (V \times W)$ such that it satisfies

Type 2: assume $\widetilde R \in F(V \times W), \widetilde B \in F(U \times W)$. We seek an interval type-2 fuzzy input $\widetilde X \in F (U \times V)$ such that it satisfies

Remark 2.2. Take a transpose of both sides of (2.19) to get ${\widetilde R^T} \circ {\widetilde X^T} = {\widetilde B^T}$. (2.19) is equivalent to (2.18), so we only need to consider the solvability of (2.18).

3.   Interval-valued logic and its matrix representation

Definition 3.1. (1)

is called the domain of interval-valued fuzzy logic, and the interval-valued fuzzy logic variable is $P \in {I_f}$. When $\alpha = [0, 0]$ $\left({or\; \alpha = [1, 1]} \right)$, $\alpha$ degenerates into a classical logic variable.

(2)

$i = 1, \cdots, k$, then ${I_k}$ is called the domain of k-valued interval-valued fuzzy logic.

(3) Mapping

is called r-ary k-valued interval-valued logic function.

If

put the different upper and lower bounds of all interval-valued fuzzy logic variables in ${I_k}$ into the ordered set $\Theta$. If $\Theta$ does not contain zero and one, it needs to add zero or one:

In order to facilitate matrix calculation, each variable in ${I^k}$ is represented as an interval vector. If ${\underline \alpha _i}{\rm{ = }}{a_m}$ $\left({1 \le m \le s}, m \in {Z^ + }\right)$ and ${\overline \alpha _i} = {a_n}$ $\left({1 \le n \le s}, n \in {Z^ + } \right)$, then the lower bound ${\underline \alpha _i}$ can be represented by vector $\delta _s^m$ and the upper bound ${\overline \alpha _i}$ can be represented by vector $\delta _s^n$. Therefore,

Similar to the proof of theorem in paper ^[23], we can obtain Theorem 3.1.

Theorem 3.1. $f$ is a r-ary k-valued interval-valued logic function, then there exists a unique structural matrix ${M_f}$, whose algebraic form is

Remark 3.1. Structure matrix is also a special interval matrix that can be used to replace $\wedge, \vee$ and $\neg$ for algebraic operations.

In the following, we give the structure matrix of $\wedge, \vee$ and $\neg$.

Let

The ordered set $\Theta$ generated by ${I_k}$ contains $s$ different elements. To simply represent the structure matrix of $\wedge, \vee$ and $\neg$, we introduce a set of s-dimensional vectors

(1) The structure matrix of $\vee$:

When s = 3, we have

(2) The structure matrix of $\wedge$:

When s = 3, we have

4.   Main results

4.1.   Decomposition of IT2 FRE

Definition 4.1. ^[20] In (2.17), the matrix constructed by the primary membership grade ${\mu _{\widetilde R}}\left({{v_j}, {w_k}} \right)$ is called primary fuzzy matrix of interval type-2 fuzzy relation, denoted as ${\widetilde R_\mu }\left({{\mu _{\widetilde R}}\left({{v_j}, { w_k}} \right)} \right)$ and abbreviated as ${\widetilde R_\mu }$:

Similarly, in (2.17), the matrix constructed by the secondary membership grade ${f_{\widetilde R}}\left({{v_j}, {w_k}} \right)$ is called secondary fuzzy matrix of interval type-2 fuzzy relation, denoted as ${\widetilde R_f}\left({{f_{\widetilde R}}\left({{v_j}, {w_k}} \right)} \right)$ and abbreviated as ${\widetilde R_f}$:

Clearly, (2.18) is composed of the primary fuzzy matrix equation and secondary fuzzy matrix equation.

Definition 4.2. The IT2 FRE (2.18) can be divided into two parts: primary fuzzy matrix equation and secondary fuzzy matrix equation.

(1) The primary fuzzy matrix equation is

where ${\widetilde A_\mu } \in I\left({{{\left[{0, 1} \right]}^{m \times n}}} \right)$, ${\widetilde B_\mu } \in I\left({{{\left[{0, 1} \right]}^{m \times p}}} \right)$, ${\widetilde X_\mu } \in I\left({{{\left[{0, 1} \right]}^{n \times p}}} \right)$ and ${\widetilde X_\mu }$ is unknown.

If

satisfies (4.1), then we call that ${\widetilde X_\mu }$ is the solution of (4.1). ${\underline X _\mu }$, ${\overline X _\mu }$ are lower and upper bound matrices of ${\widetilde X_\mu }$, respectively.

If

is a solution of (4.1), and for any solution ${\widetilde X_\mu }$ of (4.1), there is ${\widetilde X_\mu } \le {\widetilde H_\mu }$, then ${\widetilde H_\mu }$ is called the maximum solution of (4.1).

If

is a solution of (4.1), and for any solution ${\widetilde X_\mu }$ of (4.1), there is ${\widetilde X_\mu } \ge {\widetilde J_\mu }$, then ${\widetilde J_\mu }$ is called the minimal solution of (4.1).

If

is a solution of (4.1), and for any solution ${\widetilde X_\mu }$ of (4.1), as long as ${\widetilde X_\mu } \le {\widetilde Q_\mu }$ is satisfied, there is ${\widetilde X_\mu } = {\widetilde Q_\mu }$, then ${\widetilde Q_\mu }$ is called the minimum solution of (4.1).

(2) The secondary fuzzy matrix equation is

where ${\widetilde A_f} \in {{\cal M}_{m \times n}}$, ${\widetilde B_f} \in {{\cal M}_{m \times p}}$, ${\widetilde X_f} \in {{\cal M}_{n \times p}}$ and ${\widetilde X_f }$ is unknown.

The matrix ${\widetilde X_f }$ satisfying (4.2) is called the solution of this equation. In (4.2), the elements of ${\widetilde A_f}$ and ${\widetilde B_f}$ are all one, then the elements of ${\widetilde X_f }$ are all one.

4.2.   Solvability of primary fuzzy matrix equation

The primary fuzzy matrix Eq (4.1) is equivalent to

The conditions for the establishment of (4.3) are relatively difficult, so we first need to determine whether (4.1) has solutions.

Lemma 4.1. ^[24] Let

The T1 FRE $A \circ X = B$ has solutions if, and only if, ${A^T}\alpha B$ is a solution of this equation and ${A^T }\alpha B$ is the maximum solution of this equation. The $\alpha$ composition operation between fuzzy matrices is

where $({a_{ki}})\alpha ({b_{ij}}) = \left\{ \begin{array}{l} {b_{ij}}, {a_{ki}} > {b_{ij}}, \\ 1, {a_{ki}} \le {b_{ij}}. \end{array} \right.$

Theorem 4.1. If the primary fuzzy matrix Eq (4.1) has solutions then

is a solution of this equation and ${\widetilde H_\mu }$ is the maximum solution of this equation.

In (4.4),

when

When $\exists\ {{\underline h }_{ik}} > {{\overline h }_{ik}}$, we replace all elements of $\underline H$ that do not satisfy ${{\underline h }_{ik}} \le {{\overline h }_{ik}}$ with ${\overline h _{ik}}$; thus, generating a new lower bound matrix ${{{\underline H }_\mu }^\prime }$.

Proof. The primary fuzzy matrix Eq (4.1) has solutions, then T1 FREs

must have solutions. Lemma 4.1 implies that ${\underline H _\mu }$ and ${\overline H _\mu }$ are solutions of T1 FREs

respectively. ${\underline H _\mu }$ and ${\overline H _\mu }$ must exist in either of the following two cases.

(1) For $\forall\ {{\underline h }_{ij}} \le {{\overline h }_{ij}}$, we known that ${\underline H _\mu } \le {\overline H _\mu }$. ${\underline H _\mu }$ and ${\overline H _\mu }$ are solutions of T1 FREs

respectively. Hence,

satisfies (4.3) and ${\widetilde H_\mu }$ is a solution of the primary fuzzy matrix equation.

From Lemma 4.1, it follows that ${\underline H _\mu }$ and ${\overline H _\mu }$ are maximum solutions of T1 FREs

respectively. Clearly, ${\underline X_\mu } \le {\underline H_\mu }$ and ${\overline X _\mu } \le {\overline H _\mu }$, so

is the maximum solution of the primary fuzzy matrix equation.

(2) For $\exists\ {{\underline h }_{ij}} > {{\overline h }_{ij}}$, we know that the newly generated matrix is ${{{\underline H }_\mu }^\prime }$ and the matrix satisfies

According to

${{{\underline H }_\mu }^\prime }$ is a solution of T1 FRE

From Lemma 4.1, it follows that ${\overline H _\mu }$ is a solution of T1 FRE ${\overline A _\mu } \circ {\overline X _\mu } = {\overline B _\mu }$, respectively. Hence,

satisfies (4.3) and ${\widetilde H_\mu }$ is a solution of the primary fuzzy matrix equation.

From Lemma 4.1, it is known that ${\underline H _\mu }$ and ${\overline H _\mu }$ are, respectively, maximum solutions of T1 FREs

According to the requirement that ${\underline X _\mu } \le {\overline X _\mu }$, we construct a new matrix ${{{\underline H }_\mu }^\prime }$ based on ${\underline H _\mu }$. Clearly, ${\underline X_\mu } \le {{{\underline H }_\mu }^\prime }$ and ${\overline X _\mu } \le {\overline H _\mu }$, so

is the maximum solution of the primary fuzzy matrix equation.

In summary, ${\widetilde H_\mu }$ is a solution of the primary fuzzy matrix Eq (4.1) and is the maximum solution of this equation. □

If the primary fuzzy matrix Eq (4.1) has solutions, the next step is to explore how to construct parameter set solutions ${I^*}\left({{{\widetilde X}_\mu }} \right)$ and ${I_*}\left({{{\widetilde X}_\mu }} \right)$ of this equation.

First, take all the elements in ${\widetilde A_\mu}$and ${\widetilde B_\mu}$ and place the different upper and lower bounds of these elements in the the ordered set $\Theta$:

Construct an ordered interval-valued set $\Psi$ by the ordered set $\Theta$, defined as

Next, according to the order interval-valued set $\Psi$, we define two mappings necessary to construct the parameter set solution ${I^*}\left({{{\widetilde X}_\mu }} \right)$ and ${I_*}\left({{{\widetilde X}_\mu }} \right)$ of primary fuzzy matrix Eq (4.1).

Definition 4.3. Assuming $x \in {I_f}$, $\left[{{\xi _i}, {\xi _j}} \right] \in \Psi$.

(1) ${I_*}$: $\left[{\underline x, \overline x } \right] \to \Psi$ is

(2) ${I^*}$: $\left[{\underline x, \overline x } \right] \to \Psi$ is

Note that 1) When $\underline x {\rm{ = }}{\xi _i} \in \Xi, \ \overline x {\rm{ = }}{\xi _j} \in \Xi$,

2) When $\underline x \notin \Xi, \ \overline x {\rm{ = }}{\xi _j} \in \Xi$, there exists a unique $i$ such that ${\xi _i} < x < {\xi _{i + 1}}$, then

3) When $\underline x {\rm{ = }}{\xi _i} \in \Xi, \ \overline x \notin \Xi$, there exists a unique $j$ such that ${\xi _j} < x < {\xi _{j + 1}}$, then

4) When $\underline x \notin \Xi, \ \overline x \notin \Xi$, there exists a unique $i$ and $j$ such that ${\xi _i} < x < {\xi _{i + 1}}$, ${\xi _j} < x < {\xi _{j + 1}}$, then

By Definition 4.3, it is not difficult to derive the following properties.

Property 4.1. Let

then,

(1) ${I_*}({a_{ij}}) = {I^*}({a_{ij}}) = {a_{ij}}; {I_*}({b_{ik}}) = {I^*}({b_{ik}}) = {b_{ik}}.$

(2) ${I_*}({\widetilde A_\mu }) = {I^*}({\widetilde A_\mu } = {\widetilde A_\mu }; {I_*}({\widetilde B_\mu }) = {I^*}({\widetilde B_\mu }) = {\widetilde B_\mu }.$

(3) ${I_*}({\widetilde A_\mu } \circ {\widetilde X_\mu }) = {I_*}({\widetilde B_\mu }) = {\widetilde B_\mu }; {I^*}({\widetilde A_\mu } \circ {\widetilde X_\mu }) = {I^*}({\widetilde B_\mu }) = {\widetilde B_\mu }.$

(4) $\widetilde X_\mu \le {I^*}(\widetilde X_\mu), {I_*}(\widetilde X_\mu) \le \widetilde X_\mu.$

Property 4.2. Let $x, y \in {I_f}$, ${x_i}, {y_i} \in {I_f}$, ${i = 1, \cdots, n}$, then

(1) ${I_*}\left(x \right) \vee {I_*}\left(y \right) = {I_*}\left({x \vee y} \right); \ {I^*}\left(x \right) \vee {I^*}\left(y \right) = {I^*}\left({x \vee y} \right).$

(2) ${I_*}\left(x \right) \wedge {I_*}\left(y \right) = {I_*}\left({x \wedge y} \right); \ {I^*}\left(x \right) \wedge {I^*}\left(y \right) = {I^*}\left({x \wedge y} \right).$

(3) $\mathop \vee \limits_{i = 1}^n \left[{{I_*}\left({{x_i}} \right) \wedge {I_*}\left({{y_i}} \right)} \right] = {I_*}\left[{\mathop \vee \limits_{i = 1}^n \left({{x_i} \wedge {y_i}} \right)} \right].$

(4) $\mathop \vee \limits_{i = 1}^n \left[{{I^*}\left({{x_i}} \right) \wedge {I^*}\left({{y_i}} \right)} \right] = {I^*}\left[{\mathop \vee \limits_{i = 1}^n \left({{x_i} \wedge {y_i}} \right)} \right].$

Property 4.3. Let

then,

(1) ${I_*}({\widetilde A_\mu } \circ {\widetilde X_\mu }) = {I_*}({\widetilde A_\mu }) \circ {I_*}({\widetilde X_\mu }).$

(2) ${I^{\rm{*}}}({\widetilde A_\mu } \circ {\widetilde X_\mu }) = {I^*}({\widetilde A_\mu }) \circ {I^*}({\widetilde X_\mu }).$

Theorem 4.2. ${\widetilde X_\mu }$ is a solution of the primary fuzzy matrix Eq (4.1) if, and only if, ${I^*}\left({{{\widetilde X}_\mu }} \right)$ is a solution of the primary fuzzy matrix equation.

Proof. (Necessity) Assuming that ${\widetilde X_\mu }$ is a solution of the primary fuzzy matrix equation, it is clear that ${\widetilde A_\mu } \circ {\widetilde X_\mu }{\rm{ = }}{\widetilde B_\mu }$. By Property 4.1, it follows that

According to the Property 4.3, we know that

From (4.7) we have

By the Property 4.1, it is not difficult to obtain ${I^*}\left({{{\widetilde A}_\mu }} \right){\rm{ = }}{\widetilde A_\mu }$. From (4.8) we have

Formula (4.9) shows that ${I^*}\left({{{\widetilde X}_\mu }} \right)$ is a solution of the primary fuzzy matrix equation.

(Sufficiency) Assuming that ${I^*}\left({{{\widetilde X}_\mu }} \right)$ is a solution of the primary fuzzy matrix equation, it is clear that ${\widetilde A_\mu } \circ {I^*}\left({{{\widetilde X}_\mu }} \right){\rm{ = }}{\widetilde B_\mu }$. By Property 4.1, it follows that

Using Property 2.1, we can get

Formula (4.11) shows that $\widetilde X_\mu$ is a solution of the primary fuzzy matrix equation.

Therefore, the conclusion is correct.

Similarly, ${\widetilde X_\mu }$ is a solution of the primary fuzzy matrix Eq (4.1) if, and only if, ${I_*}\left({{{\widetilde X}_\mu }} \right)$ is a solution of the primary fuzzy matrix equation.

By Theorem 4.2, we can obtain the following corollary.

Corollary 4.1. (1) The interval matrix ${\widetilde H_\mu }$ is the maximum solution of primary fuzzy matrix Eq (4.1) if, and only if, ${I^*}\left({{{\widetilde H}_\mu }} \right)$ is the maximum solution of this equation.

(2) The interval matrix ${\widetilde J_\mu }$ is the minimum solution of primary fuzzy matrix Eq (4.1) if, and only if, ${I_*}\left({{{\widetilde J}_\mu }} \right)$ is the minimum solution of this equation.

(3) The interval matrix ${\widetilde Q_\mu }$ is the minimal solution of primary fuzzy matrix Eq (4.1) if, and only if, ${I_* }\left({{{\widetilde Q}_\mu }} \right)$ is the minimal solution of this equation.

If the primary fuzzy matrix Eq (4.1) has solutions, we next explore how to obtain parameter set solutions of this equation. By Theorem 4.2, the ordered interval-valued set $\Psi$ is sufficient to inscribe the entire parameter set solutions of the primary fuzzy matrix equation.

First, the primary fuzzy matrix equation can be rewritten to

where $k = 1, \cdots, p.$

In (4.12), the kth equality is equivalent to

where $i = 1, \cdots, m.$

Second, the logical form of the primary fuzzy matrix equation is converted to algebraic form.

For simplicity of presentation, let

With the help of Theorem 3.1, the left hand side (LHS) of (4.13) can be expressed in algebraic form:

where $i = 1, \cdots, m.$

By Property 2.2, we know that

According to (4.15), (4.14) is simplified to

From Property 2.2, it follows that

According to (4.17), (4.16) is further simplified to

Repeating the process of (4.15)–(4.18), (4.14) is finally expressed as

where $i = 1, \cdots, m,$ and

then (4.19) can be simplified to

where $i = 1, \cdots, m.$

Equation (4.20) is equivalent to

where

where "*" denotes the khatri-rao product of interval matrices.

According to the above procedure, the value of the kth row of ${\widetilde X_\mu }$ can be determined. Let $k = 1, 2, \cdots, p$, and we can obtain the parameter set solutions of the primary fuzzy matrix equation.

A specific algorithm for solving all solutions of IT2 FRE (2.18) is given in the following.

Algorithm 4.1. The following steps are used to solve the solution set of IT2 FRE (2.18).

Step. 1. Decompose IT2 FRE (2.18) to construct the primary fuzzy matrix Eq (4.1).

Step. 2. Use Theorem 4.1 to determine if there are solutions to the primary fuzzy matrix equation. If the primary fuzzy matrix equation has solutions, then proceed as follows; otherwise, IT2 FRE (equ:IT2 FRE(a)) has no solution.

Step. 3. Construct an ordered set $\Theta$ from ${\widetilde A_\mu}$ and ${\widetilde B_\mu}$

We specify

The elements in ${\widetilde A_\mu}$ and ${\widetilde B_\mu}$ can be represented as vectors to facilitate algebraic operations.

Step. 4. Construct (4.12) and convert it into the form of (4.21) to solve for the parameter set solutions of $Co{l_k}\left({{{\widetilde X}_\mu }} \right).$

Step. 5. Let $k = 1, 2, \cdots, p$, and we can get all parameter set solutions of $\left({{{\widetilde X}_\mu }} \right)$. Determine the maximum and minimum (or minimal) solutions of the primary fuzzy matrix equation.

Step. 6. Finally, based on the solution set of the primary fuzzy matrix equation and secondary fuzzy matrix equation, the solution set $\widetilde X$ of IT2 FRE is constructed.

5.   Application

Consider the following IT2 FRE,

where

First, taking a transpose on both sides of (5.1), we get

By decomposing IT2 FRE (5.2), we can obtain the primary fuzzy matrix equation and the secondary fuzzy matrix equation. From Definition 4.1, we only need to solve the primary fuzzy matrix equation to obtain the solution set of IT2 FRE. The primary fuzzy matrix equation of (5.2) can be expressed as

where

Next, use Theorem 4.1 to determine if (5.3) has solutions.

According to the above calculation, ${\underline H _\mu } \le {\overline H _\mu }$, ${\widetilde H_\mu }$ is a solution of (5.3) and ${\widetilde H_\mu }$ is the maximum solution of this equation.

Let

which needs to satisfy the following logical equation.

However, solving (5.4) directly is relatively difficult, so it needs to be converted into algebraic form.

Construct the ordered set $\Theta$ based on ${\widetilde A_\mu }$ and ${\widetilde B_\mu }$:

Represent the elements in $\Theta$ as vectors

Convert (5.4) into an algebraic equation

Let

which is equivalent to

Let

This leads to

The MATLAB program provided in the literature ^[15] is improved so that it can calculate the STP of the interval matrix. (5.5) is calculated as

Solving for (5.5), we get

where $i = 41, 42, 48, \ j = 18, 19, 20, 21.$

From the values of $[\underline x_1, \overline x_1]$, there are $3 \times 4 = 12$ parameter set solutions for $Co{l_1}\left({\widetilde X}_\mu^T \right)$, two of which do not satisfy ${\underline {\rm{x}} _{i1}} \le {\overline x _{i1}}\left({i = 1, 2} \right)$; then

Assuming

we have

where the value of $L$ has been obtained in (5.5),

Solving for (5.6), we get

where $i = 29, 30, 31$, $j = 15, 16, 22, 23, 29, 30, 36.$

Depending on the value of $[\underline x_2, \overline x_2]$, it follows that $Co{l_2}\left({{{\widetilde X}_\mu^T }}\right)$ has $3 \times 7 = 21$ parameter set solutions, six of which do not satisfy ${\underline {\rm{x}} _{i2}} \le {\overline x _{i2}}\left({i = 1, 2} \right)$; then

In summary, we can conclude that:

(1) The primary fuzzy matrix Eq (5.3) has a total of $10 \times 15 = 150$ parameter set solutions.

(2) The maximum solution of this equation is

(3) The equation has no minimum solution and only two minimal solutions,

(4) Based on the maximum and minimal solutions of the primary fuzzy matrix equation, we can work out all the parameter set solutions of the primary fuzzy matrix equation.

(5) The solution set of IT2 FRE is

6.   Conclusions

This paper focused on the solution of IT2 FRE $\widetilde A \circ \widetilde X = \widetilde B$. First, the STP of interval matrices and its properties were introduced, and the matrix representation of the interval-valued logic was given. Then, the IT2 FRE was considered as the primary fuzzy matrix equation and secondary fuzzy matrix equation. The solution of secondary fuzzy matrix is known, so only the primary fuzzy matrix equation needs to be solved. Moreover, the solvability of the primary matrix equation was studied, and a specific algorithm for solving IT2 FREs based on the STP of interval matrices was given. Finally, a numerical example was given to verify the effectiveness of the proposed method.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported in part by the Research Fund for the Taishan Scholar Project of Shandong Province of China under Grant tstp20221103, and in part by the National Natural Science Foundation of China under Grant 62273201.

Conflict of interest

The authors declare that there are no conflicts of interest in this paper.