Robustness of Interval-Valued Intuitionistic Fuzzy Reasoning Quintuple Implication Method

The interval-valued intuitionistic fuzzy quintuple implication algorithm, as an extension of the fuzzy reasoning algorithm, may better characterize and deal with uncertainty in the reasoning, but how to select distance measure and analyze the algorithm’s robustness is an important and unsolved topic. In this paper, a novel distance measure of interval-valued intuitionistic fuzzy sets is constructed based on interval-valued intuitionistic fuzzy biresiduum similarity. The unified form of the conclusion about the robustness of interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm for interval-valued intuitionistic fuzzy modus ponens(IVIFMP) and interval-valued intuitionistic fuzzy modus tollens(IVIFMT) is obtained. Especially, the robustness of the interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm based on <inline-formula> <tex-math notation="LaTeX">$G\ddot {o}del$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$Lukasiewicz$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$Goguen$ </tex-math></inline-formula> implication operators is presented. An application example and experiment are offered to demonstrate the validity of the obtained conclusion. Furthermore, the new distance metric is compared to traditional distances, and its benefits and limits are discussed. The results show that our approach to research the robustness is simpler and more representative, and the robustness of the algorithm based on other implication operators can be obtained by simple substitution.


I. INTRODUCTION
Fuzzy reasoning is the theoretical foundation of fuzzy control technology [1]. Fuzzy Modus Ponens (FMP) and Fuzzy Modus Tollens (FMT) are the two most fundamental forms of fuzzy reasoning [2]. Professor Zadeh, the father of fuzzy theory, presented the compositional rule of inference (CRI) algorithm to handle the problem of FMP and FMT [3], which has been widely utilized in time series forecasting, image processing, and target detection [4]- [6]. However, the logic of the CRI algorithm is flawed. The full implication Triple I algorithm and the reverse Triple I algorithm [7], [8] substituted the conjunction operation in the CRI method with the implication operator, which improved the logic in the process of reasoning. On the other hand, the Triple I and CRI methods have several restrictions that may result in deceptive computation results in some rare instances [9]. The fuzzy The associate editor coordinating the review of this manuscript and approving it for publication was Shun-Feng Su . reasoning quintuple implication algorithm, proposed in [9], successfully solved the above limitations. The robustness and approximation of the fuzzy reasoning quintuple implication method were explored in [10]. The interval-valued fuzzy reasoning approach has been the subject of several research investigations [11]- [13].
Intuitionistic fuzzy sets (IFSs) and interval-valued intuitionistic fuzzy sets (IVIFSs), both of which are extensions of fuzzy sets, can retain fuzzy information better than fuzzy sets and have been successfully applied in the fields of cluster analysis, multi-attribute decision-making, and pattern recognition [14]- [18]. To measure uncertainty in multi-attribute decision-making, Liu and Jiang [19] converted IVIFSs to three interval vectors and proposed a new distance measure of IVIFSs based on the distance of interval number. Verma and Merigo [20] proposed a cosine similarity measure and used it to solve real-world decision problems with interval-valued intuitionistic fuzzy information. Because of the diversity of intuitionistic fuzzy implication operators and triangular modules [21], [22], fuzzy reasoning is further extended to intuitionistic fuzzy reasoning [23]- [25]. Quintuple implication principle (QIP) and triple implication principle (TIP) on IVIFSs have been presented by Jin et al. [26], which is the extension of fuzzy reasoning quintuple implication algorithm on IVIFSs. Interval-valued Intuitionistic Fuzzy Modus Ponens (IVIFMP) and Interval-valued Intuitionistic Fuzzy Modus Ponens (IVIFMT) are two reasoning forms of interval-valued intuitionistic fuzzy reasoning.
In the fuzzy reasoning system, because the selection of membership function types and parameters is subjective, the rule and new input will deviate from ideal values to some extent, resulting in a deviation of reasoning output. If the reasoning system is sensitive to deviation, the reasoning output will most likely be erroneous. Therefore, the sensitivity of the fuzzy reasoning system to the perturbation in rules and input, referred to as the system's robustness, is an important indicator for assessing the applicability of the fuzzy reasoning system. It is meaningful to quantify the robustness mathematically [27]. The distance measure is an important tool for investigating the robustness of fuzzy reasoning algorithms. Hui et al. [28] employed the natural distance of the IFSs to define the sensitivity and analyzed the robustness of the intuitionistic fuzzy reasoning (1,2,2)-α type universal triple I algorithm. Zhou and Luo [29] used the normalized Minkowski distance and Minkowski inequality to assess the robustness of interval-valued fuzzy reasoning quintuple implication algorithm. In order to further stress the relevance between fuzzy reasoning and implication operator, Jing et al. [30] used biresiduum similarity as a disturbance parameter to analyze the robustness of the intuitionistic fuzzy reasoning reverse triple I algorithm.
Interval-valued intuitionistic fuzzy reasoning is an extension of fuzzy reasoning. Compared with fuzzy reasoning and intuitionistic fuzzy reasoning, it can better represent vague information. Nevertheless, the robustness of interval-valued intuitionistic fuzzy reasoning has not been studied because of its structural complexity and the difficulty in distance measure selection. It is feasible to use the traditional distance to analyze the robustness of fuzzy reasoning algorithm based on implication operator, but the process is complicated, especially the robustness of the interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm. In addition, the unified form of the conclusion about robustness under distinct implication operators is not available. A novel distance measure of IVIFSs, which is an interval-valued intuitionistic fuzzy number, is constructed to address these issues, and the unified form of the conclusion concerning the robustness of interval-valued intuitionistic fuzzy reasoning quintuple implication is obtained in this paper. The main work and contributions of this thesis we have done are as follows: i) The fuzzy biresiduum is extended to interval-valued intuitionistic fuzzy biresiduum, which is utilized to define the similarity of IVIFSs, and its property is analyzed.
ii) A novel distance measure of IVIFSs is constructed by the interval-valued intuitionistic fuzzy biresiduum, and the robustness of the interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm is investigated. Especially, the robustness of the algorithm based on Gödel, Lukasiewicz, and Goguen implication operators is given.
iii) An example and experiment are provided to validate the correctness of our conclusion. Furthermore, the proposed distance measure is compared to the traditional distance measure, and its advantages and drawbacks are discussed.
This paper is organized as follows: Section 2 reviews some conclusions and concepts that can be used in this paper; Section 3 defines interval-valued intuitionistic fuzzy biresiduum and constructs a new distance; Section 4 discusses the robustness of the interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm; Section 5, a computational experiment is provided to demonstrate the correctness of our conclusions, and the advantages and limitations of our method are analyzed. The final section concludes the conclusion and the future research.
Definition 1 [26]: Let X be a non-empty universe, and an interval-valued intuitionistic fuzzy set A on X can be expressed in the following form: and A f (x) denote the range of membership and the non-membership, respectively. If is tenable, the interval-valued intuitionistic fuzzy set A (x) is further degraded into the ordinary fuzzy set. In this paper, the interval-valued intuitionistic fuzzy set on X is denoted as IVIFS(X ).
Definition 3 [26]: , two operations ⊗ SL and ⊕ SL can be defined on SL as follows:

Example 1:
The three classical left-continuous t-norm ⊗ and t-cnorm ⊕ and their corresponding interval-valued intuitionistic t-norm ⊗ SL and t-cnorm ⊕ SL are as follows: (1) Let a, b ∈ [0, 1], the Gödel t-norm ⊗ G and t-cnorm ⊕ G have the following form: the Gödel interval-valued intuitionistic t-norm ⊗ G-SL and tcnorm ⊕ G-SL can be obtained from Definition 3: (2) Let a, b ∈ [0, 1], the Lukasiewicz t-norm ⊗ L and tcnorm ⊕ L have the following form: the Lukasiewicz interval-valued intuitionistic t-norm ⊗ L-SL and t-cnorm ⊕ L-SL can be obtained from Definition 3: (3) Let a, b ∈ [0, 1], the Goguen t-norm ⊗ GO and t-cnorm ⊕ GO have the following form: ) ∈ SL, the Goguen interval-valued intuitionistic t-norm ⊗ GO-SL and t-cnorm ⊕ GO-SL can be obtained from Definition 3: Theorem 1 [26]: Let ⊗ SL be an interval-valued intuitionistic t-norm derived from a left-continuous t-norm⊗. Then there exists an operation → SL on SL such that and → SL can be expressed as follows: Proposition 2 [26]: let ⊗ SL be an interval-valued intuitionistic t-norm derived from a left-continuous t-norm⊗ and (⊗ SL → SL ) be an interval-valued intuitionistic adjoint pair.
Theorem 2 [26]: ) ∈ SL and → SL be an interval-valued intuitionistic implication derived from a left-continuous tnorm ⊗. Besides, →, ⊕ and are associated operator of ⊗, we have that: Theorem 3 [26]: Suppose → SL is the residual intervalvalued intuitionistic implication on SL derived from a leftcontinuous t-norm ⊗, then the quintuple implication solution B * (y) for IVIFMP is the smallest element on IVIFS (Y ) satisfying And the solution B * (y) has the following form: Theorem 4 [26]: Suppose → SL is the residual intervalvalued intuitionistic implication on SL derived from a leftcontinuous t-norm ⊗, then the quintuple implication solution A * (x) for IVIFMT is the smallest element on IVIFS (X ) satisfying And the solution A * (x) has the following form: Definition 4 [30]: Let → be a regular implication on [0,1], Proposition 3 [30]: Let → be the residual implication on [0, 1] derived from a left-continuous t-norm ⊗, ∀a, b, c, d ∈ [0, 1], the following properties hold: Proposition 4 [30]: Suppose X be a non-empty universe, f and g are both functions on [0, 1], then some relations are satisfied: ( ) .

III. THE SIMILARITY BETWEEN INTERVAL-VALUED INTUITIONISTIC FUZZY SETS
In this section, the similarity of interval-valued intuitionistic fuzzy sets is constructed using biresiduum, and the operation relation of the interval-valued intuitionistic fuzzy biresiduum is analyzed.
and call it as interval-valued intuitionistic fuzzy biresiduum.
, the following properties are satisfied: According to Proposition 3 and Theorem 2, we have that: To facilitate the analysis of the relationship between E and F, and the following four parts of E are deduced and analyzed respectively: (i) The part e 1 in E.
The relationship between e 2 in E and f 2 in F is e 2 ≤ f 2 , and the proof is similar to (i). VOLUME 10, 2022 (iii) The part e 3 in E.
The relationship between e 4 in E and f 4 in F is e 4 ≥ f 4 , and the proof is similar to (iii).
According to the results of (i)-(iv),), we have that Lemma 3: Let X be a non-empty universe, ∀A (x) , A (x) ∈ IVIFS (X ), they have the following properties: ). Proof: The proof of Lemma 3 is similar to Lemma 2.

IV. THE ROBUSTNESS OF INTERVAL-VALUED INTUITIONISTIC FUZZY REASONING QUINTUPLE IMPLICATION METHOD
The distance measure is an important tool to analyze the robustness of fuzzy reasoning. This part constructs a novel distance measure d SL and analyzes the robustness of the interval-valued intuitionistic fuzzy reasoning quintuple implication method. Fig.1 depicts the analytic process about the robustness of the algorithm for IVIFMP. When the new input and output become B * (y) (B * (y)) and A * (x) (A * (x)) respectively, the procedure in Fig.1 becomes the analysis process of the algorithm for IVIFMT. Definition 7: Suppose X is a non-empty universe, A(x), A (x) ∈ IVIFS(X ), σ ∈ SL, ∀x ∈ X , S SL (A, A ) is the similarity between A(x) and A (x), σ is the largest IVIFS Due to the similarity of fuzzy sets having the opposite meaning to distance measure, and they are a pair of dual concepts [31], we can further define a similar dual distance between interval-valued intuitionistic fuzzy sets.
Definition 8: Suppose X is a non-empty universe, A(x), A (x) ∈ IVIFS(X ), ∀x ∈ X , the novel distance between A(x) and A (x) is that According to property (1) in Proposition 2, it is easy to have . So, we have that: If (A, A ) represents the degree belonging to ''large'' of the distance between A and A . The higher the similarity between A and A , the smaller the distance, that is, the smaller the deviation.
Proof: According to Lemma 2 and Lemma 3, we have In Corollary 1, we can get that when δ 1 , δ 2 , and δ 2 are all close to 0, the δ IVIFMP is also close to 0.

Corollary 7:
. Next, we provide an example of interval-valued intuitionistic fuzzy reasoning with a single rule to prove the robustness in Corollary 4.
Example 2: Suppose there is a patient in the hospital who needs to be diagnosed based on four symptoms: body temperature (represented as x 1 ), headache (represented as x 2 ), stomachache (represented as x 3 ), cough (represented as x 4 ), and chest tightness (represented as x 5 ). Let X be symptoms set and Y be disease set. Let (⊗ SL , → SL ) = (⊗ L−SL , → L−SL ), the next step is to compute B * (y) by Theorem 3, where And we can get that:  The new input is that: (1) Next, we analyze the deviation of the input and rule between cases (1) and (2). By definition 7, the similarity and distance of B * (y) and B * (y) can be obtained as follows: Obviously, according to definition 2, the following relationship can be obtained:

V. EXPERIMENT AND DISCUSSION
To further verify the correctness of the robustness conclusions we obtained, this section calculates the solution of the interval-valued intuitionistic quintuple implication algorithm for IVIFMP with different degrees of perturbation and analyzes the relationship between d SL (B * , B * ) and δ IVIFMP , where B * is the ideal solution and B * is the perturbed solution. In addition, the distance measure we proposed is compared with the traditional distance, and the advantages and limits of our analysis approach are discussed.

A. THE REASONING IN THE IDEAL CASE
In the fusion of MRI and CT medical images, the brighter areas of the two images will be extended into the fusion image. The image pixels can be divided into three gray levels based on the gray value of the image: Dark, Normal, and Bright, abbreviated as D, N , and B. The degree of image pixels belonging to the gray level (D, N , or B) can be represented by the IVIFS u − n , u + n , v − n , v + n , which can be obtained using the Gaussian membership function as follows: n represents one of the gray levels D, N , and B, and c n is the mean term of the gray level n. In the ideal scenario, set the variance a to 0.5 and the mean parameter c n of the IVIFSs corresponding to the three gray levels (D, N , and B) to 0, 0.5, and 1, respectively, for input and output.
Brighter pixels have higher fusion priority in MRI and CT image fusion [32], so the rule selected for this experiment is The rule is concreted using the upper and lower bounds of the membership and non-membership functions to simplify calculations. If an expert gives a judgment based on his own experience: ''if the normalized gray value in MRI is 0.2, and the normalized gray value in CT is 0.6, then the normalized gray value of the output is 0.55''. we substitute the expert experience into the membership and non-membership functions and obtain that  The robustness of the reasoning algorithm obtained by different distance measures is different. Thus, the distance measure d L−SL we proposed is compared to the traditional measure (the continuous intuitionistic fuzzyordered weighted distance(C-IFOWD) d C−IFOWD , Hamming distance d H , and Euclidean distance d E , which are all introduced in [19]), and the relationship of the robustness they obtained is analyzed.
We know that our distance is an interval-valued intuitionistic fuzzy number, and the traditional distance is an ordinary fuzzy number from the definition of distance. From another perspective, ordinary fuzzy numbers can also be expressed as interval-valued intuitionistic fuzzy numbers. For example, if d = 0.3, d can also be expressed as ([0.3, 0.3],[0.7, 0.7]), so the traditional distance measure can also be regarded as a special interval-valued intuitionistic fuzzy distance. Based on this, we compare our distance with the traditional distance in four dimensions ([u − , u + ], [v − , v + ]), which is the lower bound of membership degree, the upper bound of membership degree, the lower bound of non-membership degree and the upper bound of non-membership degree respectively.
The results of the comparison are shown in Fig.3. From  Fig.3, the distance measure we proposed has the same tendency as the traditional distance measure, and it is more sensitive to perturbation. Hence, our distance measure is more suitable for measuring the difference between IVIFSs. Respectively, Figure.3(a) and Figure.
L−IVIFMP can be obtained from Figure.3(c) and Figure.3(d). According to the partial order in IVIFSs, we can get d H ≤ δ L−IVIFMP , which illustrates that our conclusion in Corollary 4 is correct, and the C-IFOWD, Hamming distance, and Euclidean distance also satisfies our conclusion. The robustness we obtained can represent the robustness based on C-IFOWD, Hamming distance, and Euclidean distance.
In summary, our approach to research the robustness has the following advantages: (i) The distance measure we presented is composed of the interval-valued intuitionistic fuzzy implication operator (IVIFIO), consistent with the interval-valued intuitionistic fuzzy reasoning algorithm. The property of IVIFIO makes our analytical method easier than the method based on the traditional distance measure.
(ii) The conclusion we obtained is a unified form, and the robustness of the reasoning algorithms based on the other residual implication operators can be obtained by simple substitution (the robustness based on Lukasiewicz, Gödel, and Goguen operators are obtained in our work).
(iii) The proposed distance measure is more sensitive to perturbation than other traditional distance measures. It is more suitable for measuring the difference between IVIFSs. Furthermore, the robustness of the interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm we obtained is more representative than the traditional distance measure.
However, our method also has a limitation: the method is too dependent on implication operators, and the algorithm's robustness based on complicated implication operators is difficult to obtain, and its structure is also complex, as in Corollary 5 and Corollary 8.

VI. CONCLUSION
This paper proposes a new distance measure constructed by interval-valued intuitionistic fuzzy biresiduum. Then, based on the properties of interval-valued intuitionistic fuzzy biresiduum, the robustness of the interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm is analyzed using the new distance measure. The analysis process is uncomplicated and easy to understand. The corollary demonstrates that the interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm for IVIFMP and the algorithm for IVIFMT has different robustness. Especially, the robustness of interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm based on Lukasiewicz, Gödel and Goguen implication operators are obtained. The interval-valued intuitionistic fuzzy reasoning quintuple implication algorithms based on other implication operators can derive their robustness through our unified conclusion. One numerical example and experiment are given to demonstrate the correctness of our conclusion. the proposed distance measure is compared with other traditional distance measures in the end and the distance measure is more sensitive to disturbance than traditional distance. The algorithm's robustness based on the distance metric we represent is representative.
In future research, we shall try to extend the fuzzy reasoning quintuple implication algorithm to Pythagorean fuzzy sets and interval-valued Pythagorean fuzzy sets [33]- [35]. The distance measure proposed in this paper could also be extended, and the extended distance measure could be used to analyze the robustness of the Pythagorean fuzzy reasoning quintuple implication algorithm or the interval-valued Pythagorean fuzzy reasoning quintuple implication algorithm. In addition, our another future research direction is to apply fuzzy reasoning algorithm based on different fuzzy sets and implication operators to group decision making. LIXIANG LEI was born in Hunan, China, in 1998. She is currently pursuing the master's degree in electronic information with the College of Information Science and Engineering, Jishou University, China. Her research interests include pattern recognition theory and application.