Robustness of interval-valued universal triple I algorithms 1

Abstract

In this paper, the interval-valued triple I algorithms based on two inference models, i.e. fuzzy modus ponens and fuzzy modus tollens, are extended to the (1,2,1)-type universal triple I algorithms. The corresponding (1,2,1)-type universal triple I solutions are given. Moreover, the robustness of interval-valued (1,2,1)-type universal triple I algorithms are studied. As the corollaries of the main results, the sensitivity of [α, β]-(1,2,1)-type universal triple I solutions based on interval-valued Lukasiewicz implication and $R_{0}$ implication are given. In particularly, the sensitivity of α-(1,2,1)-type universal triple I methods based on classical sets are given.

Keywords

Interval-valued fuzzy inference (1,2,1)-type universal triple I method the sensitivity of the interval-valued fuzzy inference robustness

1 Introduction

It is well known that the most fundamental forms of fuzzy reasoning are fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT), which can be respectively expressed as follows [25, 28]:

Given the input “x is A^*” and fuzzy rule “if x is A then y is B”, try to deduce a reasonable output “y is B^*”, fuzzy modus ponens (FMP);

Given the input “y is B^*” and fuzzy rule “if x is A then y is B”, try to deduce a reasonable output “x is A^*”, fuzzy modus ponens (FMT);

In the above models, A and A^* belong to fuzzy sets $F (X)$ in the non-empty set X, B and B^* belong to fuzzy sets $F (Y)$ in the non-empty set Y.

To solve fuzzy reasoning FMP problem, the most basic method is Zadeh’s compositional rule of inference (CRI for short) [27], it has been applied successfully many fields. But the CRI method has some disadvantages [22 , 25], for example, the composition operation in the CRI method is short of clear logic meaning, and the CRI method is not reductive. As an alternative for CRI method, wang [22] proposed triple I method with full inference rule that utilized the implication operator in every step of the reasoning. This method may bring fuzzy reasoning within the framework of logical semantic implication [23], and it can be considered as a reasonable complement for the CRI method. Consequently, a considerable number of studies on the triple I method have been reported in recent years. The unified triple I algorithms based on regular implications and normal implications have been established by Wang and Fu [24]. For all residuated implications induced by left continuous t-norms, unified full implication triple I algorithms and unified full implication α-triple I algorithms of fuzzy reasoning were constructed by Pei [20]. The parametric triple I method by the combination of Schweizer-Sklar operators and triple I principles for fuzzy reasoning was investigated by Luo and Yao [15]. The robustness of full implication inference methods was studied by Dai et al. [3].

Fuzzy reasoning based on differently implication operator was proposed by Li [12]. Differently implication universal (1,2,2) type triple I method was investigated by Tang and Liu [21]. But the differently implication universal (1,2,2) type triple I method is short of application background. So differently implicational universal (1,2,1) type triple I algorithms and differently implication universal (1,2,1) type α-triple I algorithms was investigated by Luo [13, 14].

Since type-2 fuzzy set introduced by Zadeh [26] can provide us with more design degrees of freedom, type-2 fuzzy reasoning has been generally acknowledged as being advantageous and potential in uncertainty modeling [10, 17]. As a special type-2 fuzzy set, interval-valued fuzzy set (of which traditional [0, 1]-valued membership degrees are replaced by intervals in [0, 1]) can not only effectively reduce the loss of fuzzy information but also reflect the vagueness and uncertainty in information processing. Furthermore, interval-valued fuzzy set is considerably easier to handle than type-2 fuzzy set in practical applications. A constructive method for the definition of interval-valued fuzzy implication operators was introduced by C. Alcalde [1].

Due to the advantages of interval-valued fuzzy set, in recent years, researchers have been investigated the properties. Compositional rule of inference (CRI) to the case of interval-valued fuzzy set was first extended and the robustness of interval-valued fuzzy reasoning based on ICRI was discussed by Li et al. [11]. Full implication triple I algorithms was extended to the case of interval-valued fuzzy set by Luo [16]. And the robustness of full implication algorithms based on interval-valued fuzzy inference was investigated by Luo [16]. In this paper, the interval-valued full implication triple I algorithms is extended to the case of interval-valued [α, β]-(1,2,1)-type universal triple I algorithms.

The rest of this paper is organized as follows: Section 2 recalls some basic concepts for interval-valued fuzzy set. Then, in Section 3, the interval-valued universal triple I principle for fuzzy inference based on two inference models, i.e. fuzzy modus ponens and fuzzy modus tollens, is extended to the case of (1,2,1)-type, and the corresponding (1,2,1)-type universal triple I solutions are given. In Section 4, the robustness of interval-valued (1,2,1)-type universal triple I algorithms is investigated. Section 5 concludes this paper.

2 Preliminaries

It is well known that fuzzy logic connectives involve fuzzy conjunction, fuzzy disjunction, fuzzy complement and fuzzy implication. t-norms, s-norms and negations are usually utilized to represent fuzzy conjunction, disjunction and fuzzy complement, respectively. Fuzzy implication operators are a generalization of the well-known classical implication operator. In this section, we first recall the concepts of fuzzy implications, t-norms and residuated implication operators defined on interval-valued. Let SI = {[x, y] ∣ x ≤ y ; x, y ∈ [0, 1]} . An ordering on SI as [a, b] ≤ [c, d] if a ≤ c and b ≤ d is called component-wiseorder or Kulisch-Miranker order [4]. ‘∧’ is defined as [a, b] ∧ [c, d] = [a ∧ c, b ∧ d]. ‘∨’ is defined as [a, b] ∨ [c, d] = [a ∨ c, b ∨ d]. It is easy to verify that the ordering just defined is a partially ordering on SI. Furthermore, take [a, b] ∧ [c, d] = [a, b] iff [a, b] ≤ [c, d] and [a, b] ∨ [c, d] = [c, d] iff [a, b] ≤ [c, d] . We can verify that the algebraic structure (SI, ∧ , ∨ , [0, 0] , [1, 1]) is a complete, bounded and distributive lattice.

Definition 2.1. [3] An associative, commutative and isotone operation $T : SI \times SI \to SI$ is called a t-norm on SI if it satisfies $T ([x, y], [1, 1]) = [x, y]$ for any [x, y] ∈ SI .

Definition 2.2. [7] Let T be a t-norm defined on the interval [0, 1] , the associated t-norm on SI is defined as follows: $T : SI \times SI \to SI,$ where $T ([a, b], [c, d]) = [T (a, c), T (b, d)] .$ It is immediate to verify that the operation defined in this way fulfills the properties of the t-norms defined on SI. The associated t-norm $T$ on SI is called left continuous, if T is left continuous t-norm on the interval [0, 1] .

The associated t-norm $T$ on SI obtained is an extension of the t-norm on [0, 1], since if we identify each element a ∈ [0, 1] with the interval [a, a] : $T ([a, a], [b, b]) = [T (a, b), T (a, b)] \forall a, b \in [0, 1]$

Definition 2.3. [1] An interval-valued fuzzy implication → is a mapping from SI × SI to SI which is decreasing in its first component and increasing in its second component, and which satisfies [0, 0] → [0, 0] = [1, 1] , [0, 0] → [1, 1] = [1, 1] , [1, 1] → [1, 1] = [1, 1] , [1, 1] → [0, 0] = [0, 0] .

Definition 2.4. [1] For every [a, b] , [c, d] ∈ SI, an interval-valued $R$ -implication is defined by:

$R ([a, b], [c, d]) = ⋁ {[x, y] \in SI ∣ T ([a, b], [x, y]) \leq [c, d]}$ where $T$ is a t-norm on SI.

Lemma 2.1. [1] Given $T$ the t-norm defined on SI associated to the t-norm T, every residuated implication between intervals $R$ associated to the t-norm $T$ has the form: $R ([a, b], [c, d]) = [R_{T} (a, c) \land R_{T} (b, d), R_{T} (b, d)]$ where R_T is the residuated implication associated to the t-norm T on [0, 1] .

The residuated implication between intervals $R$ associated to the t-norm $T$ is an extension of the residuated implication on [0, 1], since if we identify each element a ∈ [0, 1] with the interval [a, a] :

$\begin{matrix} R ([a, a], [b, b]) & = & [R_{T} (a, b) \land R_{T} (a, b), R_{T} (a, b)] \\ = & [R_{T} (a, b), R_{T} (a, b)] . \end{matrix}$

Example 1. [9, 16] If T is a left-continuous t-norm on ([0, 1], min,max), α ∈ [0, 1] and the mapping $T_{T, α}$ [5] is defined, for [x₁, x₂] , [y₁, y₂] ∈ SI, by the formula

$T_{T, α} ([x_{1}, x_{2}], [y_{1}, y_{2}]) = [T (x_{1}, y_{1}), \max (T (α, T (x_{2}, y_{2})), T (x_{1}, y_{2}), T (x_{2}, y_{1}))],$

then $(SI, ⊓, ⊔, T_{T, α}, R_{T_{T, α}}, [0, 0], [1, 1])$ is a standard IVRL, in which the residual implicator $R_{T_{T, α}}$ of $T_{T, α}$ is given by

$\begin{matrix} R_{T_{T, α}} ([x_{1}, x_{2}], [y_{1}, y_{2}]) \\ = & [\min (R_{T} (x_{1}, y_{1}), R_{T} (x_{2}, y_{2})), \min (R_{T} \\ (T (x_{2}, α), y_{2}), R_{T} (x_{1}, y_{2}))] . \end{matrix}$ where R_T is the residuated implication of the t-norm T on [0, 1].

This construction can be easily generalized for an arbitrary residuated lattice (L, ⊓ , ⊔ , ∗ , ⇒ , 0, 1) and its triangularization, similarly as in [8]. If α = 1, $T_{T, α}$ is called t-representable, and if α = 0, $T_{T, α}$ is called pseudo t-representable [6, 9].

Remark 2.1. (1) (See [9, 16]) If α = 1, we obtain t-representable t-norms on SI: $T_{T, 1} ([x_{1}, x_{2}], [y_{1}, y_{2}]) = [T (x_{1}, y_{1}), T (x_{2}, y_{2})] .$ Then t-representable t-norms $T_{T, 1} ([x_{1}, x_{2}], [y_{1}, y_{2}])$ is the associated t-norm on SI in Definition 2.2, we denote $T_{T, 1} ([x_{1}, x_{2}], [y_{1}, y_{2}])$ by $T ([x_{1}, x_{2}], [y_{1}, y_{2}])$ .

(2) If α = 1, we obtain residual implicator of t-representable t-norms on SI: $\begin{matrix} R_{T_{T, 1}} ([x_{1}, x_{2}], [y_{1}, y_{2}]) \\ = & [\min (R_{T} (x_{1}, y_{1}), R_{T} (x_{2}, y_{2})), R_{T} (x_{2}, y_{2}))] . \end{matrix}$ Then residual implicator $R_{T_{T, 1}} ([x_{1}, x_{2}], [y_{1}, y_{2}])$ is residuated implication between intervals $R$ associated to the t-norm $T$ in Lemma 2.1, we denote $R_{T_{T, 1}} ([x_{1}, x_{2}], [y_{1}, y_{2}])$ by $R ([x_{1}, x_{2}], [y_{1}, y_{2}])$ .

(3) A t-representable t-norm $T$ is residuated iff $T$ is left-continuous.

In this article, we suppose that $T$ is left-continuous t-representable t-norm on SI, $R$ is residuated implication induced by left-continuous t-representable t-norm on SI.

Lemma 2.2. [16] Suppose $R$ is an interval-valued residuated implication reduced by a left continuous t-norm $T$ on SI, Then $\begin{matrix} T ([x, y], [x_{1}, y_{1}]) \leq [x_{2}, y_{2}] \\ \Leftrightarrow [x, y] \leq R ([x_{1}, y_{1}], [x_{2}, y_{2}]) . \end{matrix}$

Lemma 2.3. [16] Let $R$ be residuated implication between intervals associated to the t-norm $T$ , then $R ([a, b], [c, d]) = [1, 1]$

iff [a, b] ≤ [c, d] , ∀ [a, b] , [c, d] ∈ SI .

Lemma 2.4. [1] Suppose that $R$ is an interval-valued $R$ -implication induced by a t-norm $T$ on SI, for ∀ [a, b] , [c, d] , [a₁, b₁] , [c₁, d₁] ∈ SI, then these properties of the interval-valued $R$ -implication are true:

If [a, b] ≤ [a₁, b₁], then $R ([a_{1}, b_{1}], [c, d]) \leq R ([a, b], [c, d]);$

If [c, d] ≤ [c₁, d₁] , then $R ([a, b], [c, d]) \leq R ([a, b], [c_{1}, d_{1}]);$

$R ([0, 0], [a, b]) = [1, 1];$

$R ([a, b], [1, 1]) = [1, 1];$

$R ([1, 1], [a, b]) = [a, b];$

$R ([a, b], [a, b]) = [1, 1];$

$[c, d] \leq R ([a, b], [c, d]) .$

Lemma 2.5. [8] Suppose that $R$ is an interval-valued residuated implication induced by a t-norm $T$ on SI, for ∀ [a, b] , [c, d] , [e, f] ∈ SI, then $R (T ([a, b], [c, d]), [e, f]) = R ([a, b], R ([c, d], [e, f]))$

Example 2. [1, 16]

(1) The interval-valued Gödel implication implication and the corresponding t-norm:

$R_{G} ([a, b], [c, d]) = {\begin{matrix} [c, d], & if a > c & and b > d, \\ [c, 1], & if a > c & and b \leq d, \\ [1, 1], & if a \leq c & and b \leq d, \\ [d, d], & if a \leq c & and b > d . \end{matrix}$ $T_{G} ([a, b], [c, d]) = [a \land c, b \land d]$ (2) The interval-valued Lukasiewicz implication and the corresponding t-norm:

$R_{L} ([a, b], [c, d]) = {\begin{matrix} [(1 - a + c) \land (1 - b + d), \\ 1 - b + d], & if a > c & and b > d, \\ [1 - a + c, 1], & if a > c & and b \leq d, \\ [1, 1], & if a \leq c & and b \leq d, \\ [1 - b + d, 1 - b + d], & if a \leq c & and b > d . \end{matrix}$ $T_{L} ([a, b], [c, d]) = [0 \lor (a + c - 1), 0 \lor (b + d - 1)]$ (3) The interval-valued $R_{0}$ implication, or interval-valued nilpotent minimum implication, and the corresponding t-norm:

$ℛ_{0} ([a, b], [c, d]) = \{\begin{cases} [(a^{'} \lor c) \land (b^{'} \lor d), b^{'} \lor d], if a > c and b > d, \\ [(a^{'} \lor c) \land 1, 1], if a > c and b \leq d, \\ [1, 1], if a \leq c and b \leq d, \\ [1 \land (b^{'} \lor d), b^{'} \lor d], if a \leq c and b > d \end{cases}$ where $a^{'} = 1 - a, b^{'} = 1 - b$

$T_{0} ([a, b], [c, d]) = {\begin{matrix} [a \land c, b \land d], & if a + c > 1 & and b + d > 1 \\ [a \land c, 0], & if a + c > 1 & and b + d \leq 1 \\ [0, 0], & if a + c \leq 1 & and b + d \leq 1 \\ [0, b \land d], & if a + c \leq 1 & and b + d > 1 \end{matrix}$

Definition 2.5. (Moore metric [18]) For any [x₁, y₁] , [x₂, y₂] ∈ SI, d ([x₁, y₁] , [x₂, y₂]) = max {|x₁ - x₂|, |y₁ - y₂|} is referred as Moore metric.

Indeed, d fulfills the following conditions:

Positive definiteness. d ([x₁, y₁] , [x₂, y₂]) ≥0, d ([x₁, y₁] , [x₂, y₂]) =0 if and only if [x₁, y₁] = [x₂, y₂] ;

Symmetry. d ([x₁, y₁] , [x₂, y₂]) = d ([x₂, y₂] , [x₁, y₁]) ;

Triangle inequality. d ([x₁, y₁] , [x₃, y₃]) ≤ d ([x₁, y₁] , [x₂, y₂]) + d ([x₂, y₂] , [x₃, y₃]) .

3 The (1,2,1)-type triple I algorithms of based on interval-valued fuzzy inference

M.X. Luo defined the triple I principle based on interval-valued fuzzy inference for fuzzy modus ponens (IFMP) and fuzzy modus tollens (IFMT) [16]. In this section, the interval-valued full implication triple I principle fuzzy inference of interval-valued fuzzy set is extended to the case of [α, β]-(1, 2, 1)-type. Let X and Y be non-empty sets, SI (X) and SI (Y) respectively denote interval-valued fuzzy subsets of non-empty sets X and Y, A (x) , A^* (x) ∈ SI (X) (A (x) denoted by [A_l (x) , A_r (x)]) and B (y), B^* (y) ∈ SI (Y). Suppose the first implication operator is the same as the last one, which is a residuated implication induced by interval-valued left continuous t-norm $T_{1},$ and the middle one is a different residuated implication induced by an interval-valued t-norm $T_{2},$ then the interval-valued [α, β]-(1, 2, 1)-type triple I algorithm derived from $R_{2} (R_{1} (A (x), B (y)), R_{1} (A^{*} (x), B^{*} (y))) \geq [α, β], x \in X, y \in Y,$ where [α, β] is fixed interval-valued belong to [0, 1].

Definition 3.1. Suppose that $R_{1}, R_{2}$ are interval-valued residuated implications by the left continuous t-norms $T_{1}, T_{2}$ on SI respectively. [α, β] is an interval-valued belong to [0, 1]. A (x) , A^* (x) ∈ SI (X) , and B (y) ∈ SI (Y) . Let $B_{[α, β]} (A, B, A^{*}) = {C \in SI (Y) ∣ R_{2} (R_{1} (A (x), B (y)), R_{1} (A^{*} (x), C (y))) \geq [α, β], x \in X, y \in Y} .$

If the smallest element of the set B_[α,β] (A, B, A^*) exists (denoted by $B_{[α, β]}^{*}$ ), then it is called the interval-valued [α, β]-(1, 2, 1)-type universal triple I solution of IFMP.

Definition 3.2. Suppose that $R_{1}, R_{2}$ are interval-valued residuated implications by the left continuous t-norms $T_{1}, T_{2}$ on SI respectively, [α, β] is an interval-valued belong to [0, 1]. A (x) ∈ SI (X) , and B (y) , B^* (y)∈SI (Y) . Let $A_{[α, β]} (A, B, B^{*}) = {D \in SI (X) ∣ R_{2}$ $(R_{1} (A (x), B (y)), R_{1} (D (x), B^{*} (y))) \geq [α, β], x \in X,$ y ∈ Y} .

If the greatest element of the set A_[α,β] (A, B, A^*) exists (denoted by $A_{[α, β]}^{*}$ ), then it is called the interval-valued [α, β]-(1, 2, 1)-type universal triple I solution of IFMT.

Theorem 3.1. If $R_{1}, R_{2}$ are interval-valued residuated implications by the left continuous t-norms $T_{1}$ and $T_{2}$ on SI respectively, then the interval-valued [α, β]-(1, 2, 1)-type universal triple I solution $B_{[α, β]}^{*}$ of IFMP can be expressed as follows:

$\begin{matrix} B_{[α, β]}^{*} (y) & = & ⋁_{x \in X} {T_{1} (A^{*} (x), T_{2} (R_{1} (A (x), \\ B (y)), [α, β]))}, y \in Y . \end{matrix}$

Proof. First, we shall prove: $R_{2} (R_{1} (A (x), B (y)), R_{1} (A^{*} (x), B_{[α, β]}^{*} (y))) \geq [α, β], x \in X, y \in Y .$

In fact, it follows from the expression of $B_{[α, β]}^{*} (y)$ that $T_{1} (A^{*} (x), T_{2} (R_{1} (A (x), B (y)), [α, β])) \leq B_{[α, β]}^{*} (y), x \in X, y \in Y .$ Since $R_{1}$ is an interval-valued residuated implication by the left continuous t-norm $T_{1},$ we have $T_{2} (R_{1} (A (x), B (y)), [α, β])) \leq R_{1} (A^{*} (x), B_{[α, β]}^{*} (y)) .$ Since $R_{2}$ is an interval-valued residuated implication by the left continuous t-norm $T_{2},$ then $R_{2} (R_{1} (A (x), B (y)), R_{1} (A^{*} (x), B_{[α, β]}^{*} (y))) \geq [α, β] .$

Second, we shall show that $B_{[α, β]}^{*}$ is the smallest element of B_[α,β] . Let C (y) ∈ B_[α,β] such that $R_{2}$ $(R_{1} (A (x), B (y)), R_{1} (A^{*} (x), C (y))) \geq [α, β], x \in X,$ y ∈ Y . Since $R_{1}, R_{2}$ are interval-valued residuated implications by the left continuous t-norms $T_{1}$ and $T_{2}$ , thus $T_{1} (A^{*} (x), T_{2} (R_{1} (A (x), B (y)), [α, β])) \leq C (y), x \in X, y \in Y .$

Hence, C (y) is an upper bound of ${T_{1} (A^{*} (x), T_{2} (R_{1}$ (A (x) , B (y)) , [α, β])) ∣ x ∈ X} , y ∈ Y . Thus $B_{[α, β]}^{*} (y)$ ≤C (y) , y ∈ Y . Hence $B_{[α, β]}^{*} (y)$ is the interval-valued [α, β]-(1, 2, 1)-type universal triple I solution of IFMP.

If $R_{1} = R_{2}$ , we can easily obtain the following corollary.

Corollary 3.1. If $R_{1} = R_{2} = R,$ $R$ is the residuated implication induced by the left continuous t-norm $T$ , then interval-valued [α, β]- $R$ -type universal triple I solution $B_{[α, β]}^{*}$ of IFMP can be expressed as follows:

$\begin{matrix} B_{[α, β]}^{*} (y) = T ([α, β], \\ ⋁_{x \in X} T (A^{*} (x), R (A (x), B (y)))), y \in Y . \end{matrix}$ Specially, when [α, β] = [1, 1], we have $B^{*} (y) = ⋁_{x \in X} T (A^{*} (x), R (A (x), B (y))), y \in Y .$

When the universal SI reduces to the case of classical sets, i.e. for any A (x) = [A_l, A_r] ∈ SI, if A_l = A_r, then $A (x) \in F (x)$ . We can easily obtain the following corollary.

Corollary 3.2. [14] If R₁ and R₂ are residuated implications induced by the left-continuous t-norms T₁ and T₂, then the FMP universal α-triple I solution $B_{α}^{*}$ of (1, 2, 1) type can be expressed as follows:

$\begin{matrix} B_{α}^{*} (y) & = & ⋁_{x \in X} {T_{1} (A^{*} (x), T_{2} (R_{1} (A (x), \\ B (y)), α))}, y \in Y . \end{matrix}$

Theorem 3.2. Suppose that $R_{1}, R_{2}$ are interval-valued residuated implications by the left continuous t-norms $T_{1}, T_{2}$ on SI respectively, then the [α, β]-(1, 2, 1)-type universal triple I solution $A_{[α, β]}^{*}$ of IFMT can be expressed as follows:

$\begin{matrix} A_{[α, β]}^{*} (x) & = & ⋀_{y \in Y} {R_{1} (T_{2} (R_{1} (A (x), B (y)), [α, β]), \\ B^{*} (y))}, x \in X . \end{matrix}$

Proof. First, we shall prove:

$\begin{matrix} R_{2} (R_{1} (A (x), B (y)), R_{1} (A_{[α, β]}^{*} (x), B^{*} (y))) \\ \geq & [α, β], x \in X, y \in Y . \end{matrix}$

In fact, it follows from the expression of $A_{[α, β]}^{*} (x)$ that $A_{[α, β]}^{*} (x) \leq R_{1} (T_{2} (A (x), B (y)), [α, β]), B^{*} (y)), x \in X, y \in Y .$ Since $R_{1}$ is an interval-valued residuated implication by the left continuous t-norm $T_{1},$ we have $T_{2} (R_{1} (A (x), B (y)), [α, β])) \leq R_{1} (A_{[α, β]}^{*} (x), B^{*} (y)) .$ Since $R_{2}$ is an interval-valued residuated implication by the left continuous t-norm $T_{2},$ then $R_{2} (R_{1} (A (x), B (y)), R_{1} (A_{[α, β]}^{*} (x), B^{*} (y))) \geq [α, β] .$ Second, we shall show that $A_{[α, β]}^{*} (x)$ is the greatest element of A_[α,β] . Let D (x) ∈ A_[α,β], then $R_{2}$ $(R_{1} (A (x), B (y)), R_{1} (D (x), B^{*} (y))) \geq [α, β], x \in X,$ y ∈ Y . Since $R_{1}, R_{2}$ are interval-valued residuated implication by the left continuous t-norm $T_{1}$ and $T_{2}$ , thus $R_{1} (T_{2} (R_{1} (A (x), B (y)), [α, β]), B^{*} (y)) \geq D (x), x \in X, y \in Y .$

Hence, D (x) is an lower bound of ${R_{1} (T_{2} (R_{1} (A (x), B (y)), [α, β]), B^{*} (y)) \geq D (x) ∣ y \in Y}, x \in X .$ Thus $D (x) \leq A_{[α, β]}^{*} (x), x \in X .$ Hence $A_{[α, β]}^{*} (x)$ is the the interval-valued [α, β]-(1, 2, 1)-type universal triple I solution of IFMT.

If $R_{1} = R_{2}$ , we can easily obtain the following corollary.

Corollary 3.3. If $R_{1} = R_{2} = R,$ $R$ is the residuated implication induced by the left continuous t-norm $T$ , then interval-valued [α, β]- $R$ -type universal triple I solution $A_{[α, β]}^{*}$ of IFMT can be expressed as follows:

$\begin{matrix} A_{[α, β]}^{*} (x) & = & R ([α, β], ⋀_{y \in Y} R (R (A (x), B (y)), \\ B^{*} (y)))), y \in Y . \end{matrix}$ Specially, when [α, β] = [1, 1], we have $A^{*} (x) = ⋀_{y \in Y} R (R (A (x), B (y)), B^{*} (y))), y \in Y .$

When the universal SI reduces to the case of classical sets, we can easily obtain the following corollary.

Corollary 3.4. [14] If R₁ and R₂ are residuated implications induced by the left-continuous t-norms T₁ and T₂, then the FMT universal α-triple I solution $A_{α}^{*}$ of (1, 2, 1) type can be expressed as follows:

$\begin{matrix} A_{α}^{*} (x) & = & ⋀_{y \in Y} {R_{1} (T_{2} (R_{1} (A (x), B (y)), α), \\ B^{*} (y))}, x \in X . \end{matrix}$

4 Robustness of the interval-valued (1, 2, 1)-type triple I algorithms

Definition 4.1. [11] Let f be an n-tuple mapping from SIⁿ to SI and ɛ ∈ [0, 1]. For arbitrary [x, y] = ([x₁, y₁] , [x₂, y₂] , …, [x_n, y_n]) ∈ SIⁿ, the ɛ sensitivity of f at point [x, y] is defined by

Δ_f ([x, y] , ɛ) = ⋁ {d (f ([x, y]) , f ([x′, y′])) | [x′, y′] ∈ SIⁿ and dⁿ ([x, y] , [x′, y′]) ≤ ɛ} .

where $d^{n} ([x, y], [x^{'}, y^{'}]) = max {max_{i} ∣ x_{i} - x_{i}^{'} ∣, max_{i}$ $∣ y_{i} - y_{i}^{'} ∣} .$

Definition 4.2. [11] The maximum ɛ sensitivity of f is defined as follows: $Δ_{f} (ɛ) = ⋁_{[x, y] \in {SI}^{n}} Δ_{f} ([x, y], ɛ) .$

Definition 4.3. [11] Let f and be two n-tuple interval fuzzy connectives. We say that f is at least as robust as if ∀ɛ > 0, Furthermore, if there exists ɛ ≥ 0 such that then f is called more robust than f′.

Definition 4.4. [11] Let A and A′ be two interval-valued fuzzy sets on SI. If ∥A - A′ ∥ _∞ = ⋁ _x∈Xd (A (x) , A′ (x)) ≤ ɛ hold for all x ∈ X, then A′ is called the ɛ-perturbation of A denoted by A′ ∈ B (A, ɛ) .

Definition 4.5. Let A, A′, B, B′, A^* and A^′* be interval-valued fuzzy sets on SI. If ∥A - A′ ∥ _∞ ≤ ɛ, ∥B - B′ ∥ _∞ ≤ ɛ, ∥A^* - A^′* ∥ _∞ ≤ ɛ, and $B_{[α, β]}^{*}$ and $B^{'_{[α, β]}^{*}}$ are the interval-valued [α, β]-(1, 2, 1)-type universal triple I solutions of IFMP (A, B, A^*) and IFMP (A′, B′, A^′*) given by Theorem 3.1 respectively, then the sensitivity of the interval-valued fuzzy inference IFMP denoted as $Δ_{B_{[α, β]}^{*}}$ is defined as follows: $\begin{matrix} Δ_{B_{[α, β]}^{*}} & = & ∥ B_{[α, β]}^{*} - B^{'_{[α, β]}^{*}} ∥_{\infty} \\ = & ⋁_{y \in Y} d (B_{[α, β]}^{*} (y), B^{'_{[α, β]}^{*}} (y)) . \end{matrix}$

Definition 4.6. Let A, A′, B, B′, B^* and B^′* be interval-valued fuzzy sets on SI. If ∥A - A′ ∥ _∞ ≤ ɛ, ∥B - B′ ∥ _∞ ≤ ɛ, ∥B^* - B^′* ∥ _∞ ≤ ɛ, and $A_{[α, β]}^{*}$ and $A^{'_{[α, β]}^{*}}$ are the interval-valued [α, β]-(1, 2, 1)-type universal triple I solutions of IFMT (A, B, B^*) and IFMT (A′, B′, B^′*) given by Theorem 3.2 respectively, then the sensitivity of the interval-valued fuzzy inference IFMT denoted as $Δ_{A_{[α, β]}^{*}}$ is defined as follows:

$\begin{matrix} Δ_{A_{[α, β]}^{*}} & = & ∥ A_{[α, β]}^{*} - A^{'_{[α, β]}^{*}} ∥_{\infty} \\ = & ⋁_{y \in Y} d (A_{[α, β]}^{*} (x), A^{'_{[α, β]}^{*}} (x)) . \end{matrix}$

Lemma 4.1. For a binary interval-valued fuzzy connective f : SI × SI → SI, we have

(i) If f is any t-norm on SI, then $\begin{matrix} Δ_{f} (([x_{1}, y_{1}], [x_{2}, y_{2}]), ɛ) \\ = & (f_{l} ([x_{1}, y_{1}], [x_{2}, y_{2}]) - f_{l} ([(x_{1} - ɛ) \lor 0, \\ (y_{1} - ɛ) \lor 0], [(x_{2} - ɛ) \lor 0, (y_{2} - ɛ) \lor 0])) \\ \lor & (f_{r} ([x_{1}, y_{1}], [x_{2}, y_{2}]) - f_{r} ([(x_{1} - ɛ) \lor 0, \\ (y_{1} - ɛ) \lor 0], [(x_{2} - ɛ) \lor 0, (y_{2} - ɛ) \lor 0])) \\ \lor & (f_{l} ([(x_{1} + ɛ) \land 1, (y_{1} + ɛ) \land 1], [(x_{2} + ɛ) \land 1, \\ (y_{2} + ɛ) \land 1]) - f_{l} ([x_{1}, y_{1}], [x_{2}, y_{2}]) \\ \lor & (f_{r} ([(x_{1} + ɛ) \land 1, (y_{1} + ɛ) \land 1], [(x_{2} + ɛ) \land 1, \\ (y_{2} + ɛ) \land 1]) - f_{r} ([x_{1}, y_{1}], [x_{2}, y_{2}]) \end{matrix}$

(ii) If f is any $R$ -implication on SI, then $\begin{matrix} Δ_{f} (([x_{1}, y_{1}], [x_{2}, y_{2}]), ɛ) \\ = & (f_{l} ([x_{1}, y_{1}], [x_{2}, y_{2}]) - f_{l} ([(x_{1} + ɛ) \land 1, \\ (y_{1} + ɛ) \land 1], [(x_{2} - ɛ) \lor 0, (y_{2} - ɛ) \lor 0])) \\ \lor & (f_{r} ([x_{1}, y_{1}], [x_{2}, y_{2}]) - f_{r} ([(x_{1} + ɛ) \land 1, \\ (y_{1} + ɛ) \land 1], [(x_{2} - ɛ) \lor 0, (y_{2} - ɛ) \lor 0])) \\ \lor & (f_{l} ([(x_{1} - ɛ) \lor 0, (y_{1} - ɛ) \lor 0], [(x_{2} + ɛ) \land 1, \\ (y_{2} + ɛ) \land 1]) - f_{l} ([x_{1}, y_{1}], [x_{2}, y_{2}]) \\ \lor & (f_{r} ([(x_{1} - ɛ) \lor 0, (y_{1} - ɛ) \lor 0], [(x_{2} + ɛ) \land 1, \\ (y_{2} + ɛ) \land 1]) - f_{r} ([x_{1}, y_{1}], [x_{2}, y_{2}]) \end{matrix}$

Lemma 4.2. [11] (1) The ∧-representable t-norm on SI is the most robust t-norm, and $Δ_{T_{\land}} (ɛ) = ɛ .$

(2) The Lukasiewicz implication on SI is the most robust $R$ -implication, and $Δ_{R_{L}} (ɛ) = 2 ɛ \land 1 .$

Lemma 4.3. Suppose that ∥A - A′ ∥ _∞ ≤ ɛ, ∥B - B′ ∥ _∞ ≤ ɛ, and $R$ is an interval-valued residuated implication induced by a left continuous t-norm $T$ on SI, then

$\begin{matrix} d (T ([α, β], R (A (x), B (y))), T ([α, β], \\ R (A^{'} (x), B^{'} (y)))) \leq Δ_{T} (Δ_{R} (ɛ)) . \end{matrix}$

Specially, when [α, β] = [1, 1], we have $d (R (A (x), B (y))), R (A^{'} (x), B^{'} (y))) \leq Δ_{R} (ɛ) .$

Proof. Since ∥A - A′ ∥ _∞ ≤ ɛ and ∥B - B′ ∥ _∞ ≤ ɛ. Obviously, $d (R (A (x), B (y)), R (A^{'} (x), B^{'} (y))) \leq Δ_{R} (ɛ) .$ So we have

$\begin{matrix} d (T ([α, β], R (A (x), B (y))), T ([α, β], \\ R (A^{'} (x), B^{'} (y)))) \\ \leq & Δ_{T} (([α, β], R (A (x), B (y))), Δ_{R} (ɛ)) \\ \leq & Δ_{T} (Δ_{R} (ɛ)) . \end{matrix}$

Theorem 4.1. Suppose that ∥A - A′ ∥ _∞ ≤ ɛ, ∥B - B′ ∥ _∞ ≤ ɛ, ∥A^* - A^′* ∥ _∞ ≤ ɛ, and $B_{[α, β]}^{*} (y)$ and $B^{'_{[α, β]}^{*}} (y)$ are the interval-valued [α, β]-(1, 2, 1)-type universal triple I solutions of IFMP (A, B, A^*) and IFMP (A′, B′, A^′*) given by Theorem 3.1 respectively, then the sensitivity of the interval-valued fuzzy inference IFMP:

$\begin{matrix} Δ_{B_{[α, β]}^{*}} (ɛ) & = & ∥ B_{[α, β]}^{*} - B_{[α, β]}^{'^{*}} ∥_{\infty} \\ \leq & Δ_{T_{1}} (Δ_{T_{2}} (Δ_{R_{1}} (ɛ))) . \end{matrix}$

Proof. Since ∥A - A′ ∥ _∞ ≤ ɛ, ∥B - B′ ∥ _∞ ≤ ɛ, ∥A^* - A^′* ∥ _∞ ≤ ɛ, we have: $\begin{matrix} Δ_{B_{[α, β]}^{*}} (ɛ) = ∥ B_{[α, β]}^{*} - B_{[α, β]}^{'^{*}} ∥_{\infty} \\ = & ⋁_{y \in Y} d (B_{[α, β]}^{*} (y), B^{'_{[α, β]}^{*}} (y)) \\ = & ⋁_{y \in Y} d (⋁_{x \in X} T_{1} (A^{*} (x), T_{2} (R_{1} (A (x), B (y)), \\ [α, β])), \\ ⋁_{x \in X} T_{1} (A^{' *} (x), T_{2} (R_{1} (A^{'} (x), B^{'} (y)), \\ [α, β]))) \\ \leq & ⋁_{y \in Y} ⋁_{x \in X} d (T_{1} (A^{*} (x), T_{2} (R_{1} (A (x), B (y)), \\ [α, β])), \\ T_{1} (A^{' *} (x), T_{2} (R_{1} (A^{'} (x), B^{'} (y)), [α, β]))) \\ = & Δ_{T_{1}} ([A^{*} (x), T_{2} (R_{1} (A (x), B (y)), [α, β])], \\ Δ_{T_{2}} (Δ_{R_{1}} (ɛ))) \\ \leq & Δ_{T_{1}} (Δ_{T_{2}} (Δ_{R_{1}} (ɛ))) . \end{matrix}$ By Theorem 4.1, we can obtain the following corollaries.

Corollary 4.1. If $T_{1}$ is interval-valued Łukasiewicz t-norm on SI, $R_{1}$ is its residuum, Łukasiewicz implication $R_{L}$ , $T_{2}$ is interval-valued $T_{0}$ t-norm on SI, $R_{2}$ is its residuum implication $R_{0}$ , then $Δ_{B_{[α, β]}^{*}} (ɛ) = ɛ + Δ_{R_{L}} (ɛ) .$

Proof. Let A^∗ (x) = [x₁, y₁], A (x) = [x₂, y₂], B (y) =[x₃, y₃], $A^{'} * (x) = [x_{1}^{'}, y_{1}^{'}]$ , $A^{'} (x) = [x_{2}^{'}, y_{2}^{'}]$ , $B^{'} (y) = [x_{3}^{'}, y_{3}^{'}]$ . Suppose that – A^∗-A′∗ ∥ _∞ ≤ ɛ, ≤ɛ, ${∥B - B^{'}∥}_{\infty} \leq ε$ . By Lemma 4.1 (i), we have $\begin{matrix} d (T_{0} (R_{L} (A (x), B (y)), [α, β]), \\ T_{0} (R_{L} (A^{'} (x), B^{'} (y)), [α, β]) \\ = & | (α + (R_{L})_{l} ([x_{2}, y_{2}], [x_{3}, y_{3}]) - 1)) \\ - (α + (R_{L}) l ([x_{2}^{'}, y_{2}^{'}], [x_{3}^{'}, y_{3}^{'}]) - 1) | \\ \lor | β + (R_{L})_{r} ([β, y_{2}], [x_{3}, y_{3}]) - 1)) \\ - (β + (R_{L})_{r} ([x_{2}^{'}, y_{2}^{'}], [x_{3}^{'}, y_{3}^{'}]) - 1) | \\ \leq & | (α + (R_{L})_{l} ([x_{2}, y_{2}], [x_{3}, y_{3}]) - 1) \\ - (α + (R_{L})_{l} ([x_{2}, y_{2}], [x_{3}, y_{3}])_{l} \\ - Δ_{R_{L}} (ɛ) - 1) | \\ \lor | (β + (R_{L})_{r} ([x_{2}, y_{2}], [x_{3}, y_{3}]) - 1)) \\ - (β + (R_{L})_{r} ([x_{2}, y_{2}], [x_{3}, y_{3}]) - \\ Δ_{R_{L}} (ɛ) - 1) | \\ \leq & Δ_{R_{L}} (ɛ) \end{matrix}$ then $\begin{matrix} d (T_{1} (A^{*} (x), T_{2} (R_{1} (A (x), B (y)), [α, β])), \\ T 1 (A^{' *} (x), T_{2} (R_{1} (A^{'} (x), B^{'} (y))), [α, β]))) \\ = & | (0 \lor (x_{1} + (T_{0})_{l} (R_{L} ([x_{2}, y_{2}], [x_{3}, y_{3}]), [α, β]) \\ - 1)) - (0 \lor (x_{1}^{'} + (T_{0})_{l} (R_{L} ([x_{2}^{'}, y_{2}^{'}], [x_{3}^{'}, y_{3}^{'}]), \\ [α, β]) - 1)) | \lor | (0 \lor (y_{1} + (T_{0})_{r} (R_{L} ([x_{2}, y_{2}], \\ [x_{3}, y_{3}]), [α, β]) - 1)) - (0 \lor (y_{1}^{'} + (T_{0})_{r} (R_{L} \\ ([x_{2}, y_{2}], [x_{3}, y_{3}]), [α, β]) - 1)) | \\ \leq & | (0 \lor (x_{1} + (T_{0})_{l} (R_{L} ([x_{2}, y_{2}], [x_{3}, y_{3}]), [α, β]) \\ - 1)) - (0 \lor ((x_{1} - ɛ) + (T_{0})_{l} (R_{L} ([x_{2}, y_{2}], \\ [x_{3}, y_{3}]), [α, β]) - Δ_{R_{L}} (ɛ) - 1 | \lor | (0 \lor (y_{1} + \\ (T_{0})_{r} (R_{L} ([x_{2}, y_{2}], [x_{3}, y_{3}]), [α, β]) - 1)) \\ - (0 \lor ((y_{1} - ɛ) + (T_{0})_{r} (R_{L} ([x_{2}, y_{2}], [x_{3}, y_{3}]), \\ [α, β]) - Δ_{R_{L}} (ɛ) - 1)) | \\ \leq & ɛ + Δ_{R_{L}} (ɛ) \end{matrix}$

For interval-valued Łukasiewicz implication, for all [x₁, y₁], we can take $[x_{1}^{'}, y_{1}^{'}] = [x_{1} - ε, y_{1} - ε]$ , $R ([x_{2}, y_{2}], [x_{3}, y_{3}]) = [1, 1]$ and $R ([x_{2}^{'}, y_{2}^{'}], [x_{3}^{'},$ ensure the above equality is satisfied, i.e. $Δ_{B_{[α, β]}^{*}} (ɛ) = ɛ + Δ_{R_{L}} (ɛ) .$

Corollary 4.2. Suppose that ∥A - A′ ∥ _∞ ≤ ɛ, ∥B - B′ ∥ _∞ ≤ ɛ, ∥A^* - A^′* ∥ _∞ ≤ ɛ, and $B_{[α, β]}^{*} (y)$ and $B_{[α, β]}^{'^{*}} (y)$ are the interval-valued [α, β]- $R$ -type universal triple I solutions of IFMP (A, B, A^*) and IFMP (A′, B′, A^′*) given by Corollary 3.1 respectively, then the sensitivity of the interval-valued fuzzy inference IFMP: $Δ_{B_{[α, β]}^{*}} (ɛ) = ∥ B_{[α, β]}^{*} - B_{[α, β]}^{' *} ∥_{\infty} \leq Δ_{T} (Δ_{T} (Δ_{R} (ɛ))) .$

Specially, for Corollary 4.2, when [α, β] = [1, 1], by Lemma 4.3, we have:

Corollary 4.3. [16] Suppose that ∥A - A′ ∥ _∞ ≤ ɛ, ∥B - B′ ∥ _∞ ≤ ɛ, ∥A^* - A^′* ∥ _∞ ≤ ɛ, and B^* (y) and B^′* (y) are the interval-valued $R$ -type universal triple I solutions of IFMP (A, B, A^*) and IFMP (A′, B′, A^′*) given by Corollary 3.1 respectively, then the sensitivity of the interval-valued fuzzy inference IFMP: $Δ_{B^{*}} (ɛ) = ∥ B^{*} - B^{' *} ∥_{\infty} \leq Δ_{T} (Δ_{R} (ɛ)) .$

When the universal SI reduces to the case of classical sets, we can easily obtain the following corollary.

Corollary 4.4. Suppose that ∥A - A′ ∥ _∞ ≤ ɛ, ∥B - B′ ∥ _∞ ≤ ɛ, ∥A^* - A^′* ∥ _∞ ≤ ɛ, and $B_{α}^{*} (y)$ and $B^{'_{α}^{*}} (y)$ are the α-(1, 2, 1)-type universal triple I solutions of FMP (A, B, A^*) and FMP (A′, B′, A^′*) given by Corollary 3.2 respectively, then the sensitivity of the fuzzy inference FMP:

$Δ_{B_{α}^{*}} (ɛ) = ∥ B_{α}^{*} - B_{α}^{'^{*}} ∥_{\infty} \leq Δ_{T_{1}} (Δ_{T_{2}} (Δ_{R_{1}} (ɛ))) .$

Theorem 4.2. Suppose that ∥A - A′ ∥ _∞ ≤ ɛ, ∥B - B′ ∥ _∞ ≤ ɛ, ∥B^* - B^′* ∥ _∞ ≤ ɛ and $A_{[α, β]}^{*} (x)$ and $A^{'_{[α, β]}^{*}} (x)$ are the interval-valued [α, β]-(1, 2, 1)-type universal triple I solutions of IFMT (A, B, B^*) and IFMT (A′, B′, B^′*) given by Theorem 3.2 respectively, then the sensitivity of the interval-valued fuzzy inference IFMT:

$\begin{matrix} Δ_{A_{[α, β]}^{*}} (ɛ) & = & ∥ A_{[α, β]}^{*} - A_{[α, β]}^{'^{*}} ∥_{\infty} \\ \leq & Δ_{R_{1}} (Δ_{T_{2}} (Δ_{R_{1}} (ɛ))) . \end{matrix}$

Proof. Since ∥A - A′ ∥ _∞ ≤ ɛ, ∥B - B′ ∥ _∞ ≤ ɛ,

∥B^* - B^′* ∥ _∞ ≤ ɛ, we have

$\begin{matrix} Δ_{A_{[α, β]}^{*}} (ɛ) = ∥ A_{[α, β]}^{*} - A_{[α, β]}^{'^{*}} ∥_{\infty} \\ = & ⋁_{y \in Y} d (A_{[α, β]}^{*} (x), A^{'_{[α, β]}^{*}} (x)) \\ = & ⋁_{x \in X} d (⋀_{y \in Y} R_{1} (B^{*} (y), T_{2} (R_{1} (A (x), B (y)), \\ [α, β])), \\ ⋀_{y \in Y} R_{1} (B^{' *} (y), T_{2} (R_{1} (A^{'} (x), B^{'} (y)), \\ [α, β]))) \\ \leq & ⋁_{x \in X} ⋁_{y \in Y} d (R_{1} (B^{*} (y), T_{2} (R_{1} (A (x), B (y)), \\ [α, β])), \\ R_{1} (B^{' *} (y), T_{2} (R_{1} (A^{'} (x), B^{'} (y)), [α, β]))) \\ \leq & Δ_{R_{1}} (Δ_{T_{2}} (Δ_{R_{1}} (ɛ))) . \end{matrix}$ By Theorem 4.2, we can obtain the following corollaries.

Corollary 4.5. If $T_{1}$ is interval-valued Łukasiewicz t-norm on SI, $R_{1}$ is its residuum, Łukasiewicz implication $R_{L}$ , $T_{2}$ is interval-valued $T_{0}$ t-norm on SI, $R_{2}$ is its residuum implication $R_{0}$ , then $Δ_{A_{[α, β]}^{*}} (ɛ) = ɛ + Δ_{R_{L}} (ɛ) .$

Proof. The proofs of statement are similar to that considered above for Corollary 4.1

Corollary 4.6. Suppose that ∥A - A′ ∥ _∞ ≤ ɛ,∥B - B′ ∥ _∞ ≤ ɛ, ∥B^* - B^′* ∥ _∞ ≤ ɛ, and $A_{[α, β]}^{*} (x)$ and $A^{'_{[α, β]}^{*}} (x)$ are the interval-valued [α, β]- $R$ -type universal triple I solutions of IFMT (A, B, B^*) and IFMT (A′, B′, B^′*) given by Corollary 3.3 respectively, then the sensitivity of the interval-valued fuzzy inference IFMT: $Δ_{A_{[α, β]}^{*}} (ɛ) = ∥ A_{[α, β]}^{*} - A_{[α, β]}^{' *} ∥_{\infty} \leq Δ_{R} (Δ_{T} (Δ_{R} (ɛ))) .$

Specially, for Corollary 4.6, when [α, β] = [1, 1], by Lemma 4.3, we have

Corollary 4.7. [16] Suppose that ∥A - A′ ∥ _∞ ≤ ɛ, ∥B - B′ ∥ _∞ ≤ ɛ, ∥B^* - B^′* ∥ _∞ ≤ ɛ, and A^* (x) and A^′* (x) are the interval-valued $R$ -type universal triple I solutions of IFMT (A, B, B^*) and IFMT (A′, B′, B^′*) given by Corollary 3.3 respectively, then the sensitivity of the interval-valued fuzzy inference IFMT: $Δ_{A^{*}} (ɛ) = ∥ A^{*} - A^{' *} ∥ \infty \leq Δ_{R} (Δ_{R} (ɛ)) .$

When the universal SI reduces to the case of classical sets, we can easily obtain the following corollary.

Corollary 4.8. Suppose that ∥A - A′ ∥ _∞ ≤ ɛ, ∥B - B′ ∥ _∞ ≤ ɛ, ∥B^* - B^′* ∥ _∞ ≤ ɛ and $A_{α}^{*} (x)$ and $A^{'_{α}^{*}} (x)$ are the α-(1, 2, 1)-type universal triple I solutions of FMT (A, B, B^*) and FMT (A′, B′, B^′*) given by Corollary 3.4 respectively, then the sensitivity of the fuzzy inference FMT:

$Δ_{A_{α}^{*}} (ɛ) = ∥ A_{α}^{*} - A_{α}^{'^{*}} ∥_{\infty} \leq Δ_{R_{1}} (Δ_{T_{2}} (Δ_{R_{1}} (ɛ))) .$

5 Conclusions

In this paper, the robustness of interval-valued [α, β]-(1,2,1)-type universal triple I methods is investigated. Firstly, the interval-valued universal triple I principles for fuzzy inference based on two inference models, i.e. fuzzy modus ponens and fuzzy modus tollens, are extended to the case of [α, β]-(1,2,1)-type. In addition, the [α, β]-(1,2,1)-type triple I solutions based on interval-valued fuzzy reasoning are given. Also, the robustness of interval-valued [α, β]-(1,2,1)-type universal triple I methods are studied. As the corollaries of the main results, the sensitivity of [α, β]-(1,2,1)-type universal triple I solutions based on interval-valued Lukasiewicz implication and $R_{0}$ implication is given. In particularly, the sensitivity of α-(1,2,1)-type universal triple I methods based on classical sets is studied. It is shown that when the universal SI reduces to the case of classical sets, the interval-valued (1,2,1)-type triple I method as well as its robustness is reductive as the classical case.

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