Some preference relations based on q-rung orthopair fuzzy sets

As a generalization of intuitionistic fuzzy sets and Pythagorean fuzzy sets, $q$-rung orthopair fuzzy sets provide decision makers more flexible space in expressing their opinions. Preference relations have received widespread acceptance as an efficient tool in representing decision makers’ preference over alternatives in the decision-making process. In this paper, some new preference relations are investigated based on the $q$-rung orthopair fuzzy sets. First, a novel score function is presented for ranking $q$-rung orthopair fuzzy numbers. Second, $q$-rung orthopair fuzzy preference relation, consistent $q$-rung orthopair fuzzy preference relation, incomplete $q$-rung orthopair fuzzy preference relation, consistent incomplete $q$-rung orthopair fuzzy preference relation, and acceptable incomplete $q$-rung orthopair fuzzy preference relation are defined. In the end, based on the new score function and these preference relations, some algorithms are constructed for ranking and selection of the decision-making alternatives.     

Weighted power means of q-rung orthopair fuzzy information and their applications in multiattribute decision making

Weighted power means with weights and exponents serving as their parameters are generalizations of arithmetic means. Taking into account decision makers' flexibility in decision making, each attribute value is usually expressed by a $q$-rung orthopair fuzzy value ($q$-ROFV, $q\ge 1$), where the former indicates the support for membership, the latter support against membership, and the sum of their $q$th powers is bounded by one. In this paper, we propose the weighted power means of $q$-rung orthopair fuzzy values to enrich and flourish aggregations on $q$-ROFVs. First, the $q$-rung orthopair fuzzy weighted power mean operator is presented, and its boundedness is precisely characterized in terms of the power exponent. Then, the $q$-rung orthopair fuzzy ordered weighted power mean operator is introduced, and some of its fundamental properties are investigated in detail. Finally, a novel multiattribute decision making method is explored based on developed operators under the $q$-rung orthopair fuzzy environment. A numerical example is given to illustrate the feasibility and validity of the proposed approach, and it is shown that the power exponent is an index suggesting the degree of the optimism of decision makers.     

q-Rung orthopair fuzzy Choquet integral aggregation and its application in heterogeneous multicriteria two-sided matching decision making

In the real decision making, $q$-rung orthopair fuzzy sets ($q$-ROFSs) as a novel effective tool can depict and handle uncertain information in a broader perspective. Considering the interrelationships among the criteria, this paper extends Choquet integral to the $q$-rung orthopair fuzzy environment and further investigates its application in multicriteria two-sided matching decision making. To determine the fuzzy measures used in Choquet integral, we first define a pair of $q$-rung orthopair fuzzy entropy and cross-entropy. Then, by utilizing $\lambda$-fuzzy measure theory, we propose an entropy-based method to calculate the fuzzy measures upon criteria. Furthermore, we discuss $q$-rung orthopair fuzzy Choquet integral operator and its properties. Thus, with the aid of $q$-rung orthopair fuzzy Choquet integral, we consider the preference heterogeneity of the matching subjects and further explore the corresponding generalized model and approach for the two-sided matching. Finally, a simulated example of loan market matching is given to illustrate the validity and applicability of our proposed approach.     

The distance measures between q-rung orthopair hesitant fuzzy sets and their application in multiple criteria decision making

In this paper, we first introduce the concept of q-rung orthopair hesitant fuzzy set ($q$-ROHFS) and discuss the operational laws between any two $q$-ROHFSs. Then the distance measures between $q$-ROHFSs are proposed based on the concept of “multiple fuzzy sets”, and we develop the TOPSIS method to the proposed distance measures. The proposed distance measures not only retain the preference information expressed by $q$-ROHFSs, but also deal with the $q$-rung orthopair hesitant fuzzy decision information more objectively, In fact, the method can avoid the loss and distortion of the information in actual decision-making process. Furthermore, we give an illustrative example about the selection of energy projects to illustrate the reasonableness and effectiveness of the proposed method, which is also compared with other existing methods. Finally, we make the sensitivity analysis of the parameters in proposed distance measures about the selection of energy projects.     

Differential calculus of interval-valued q-rung orthopair fuzzy functions and their applications

The interval-valued q-rung orthopair fuzzy set (IVq-ROFS) provides an extension of Yager's q-rung orthopair fuzzy set (q-ROFS), where membership and nonmembership degrees are subsets of the closed interval $\left[0,1\right]$. In such a situation, it is more superior for decision makers to provide their judgments by intervals instead of crisp numbers due to the uncertainty and vagueness in real life. In this paper, we study the calculus theories of IVq-ROFS from the microscopic. In particular, we first introduce the elementary arithmetic of interval-valued q-rung orthopair fuzzy values (IVq-ROFVs), including addition, multiplication, and their inverse. They are the basis for analysis and calculation throughout the work. In addition, we discuss and prove in detail the operation properties and aggregation operators of IVq-ROFVs. Then, we introduce the concept of interval-valued q-rung orthopair fuzzy functions (IVq-ROFFs), which is the main research object of this paper. After that, we further discuss the continuity, derivatives and differentials of IVq-ROFFs. We also find that the derivatives of IVq-ROFFs are closely related to elasticity, which is an important concept in economics. Finally, we provide some application examples to verify the feasibility and effectiveness of the derived results.     

Linguistic Z-number fuzzy soft sets and its application on multiple attribute group decision making problems

In this study, the concept of linguistic Z-number fuzzy soft set ($\text{◂⋅▸}LZnFSS$) is proposed to describe multiple uncertainties in practical decision making problems. $\text{◂⋅▸}LZnFSS$ combines the concepts of fuzzy soft set, linguistic Z-number, and soft set, which could reflect both of the uncertainty in structure and the uncertainty in detailed evaluations. As an initial idea, the set operations on $\text{◂⋅▸}LZnFSSs$ are put forward, the properties of such operations are also discussed. With traditional soft set based decision procedure and fuzzy soft set based decision procedure, a novel linguistic Z-number fuzzy soft set based group decision procedure is developed to solve multiattribute group decision making with linguistic Z-numbers. Wherein an extended technique for order preference by similarity to ideal solution is also developed. Finally, a numerical example is shown to illustrate the practicality and effectiveness of the given method.     

Pythagorean fuzzy linguistic Muirhead mean operators and their applications to multiattribute decision-making

Pythagorean fuzzy sets, as an extension of intuitionistic fuzzy sets to deal with uncertainty, have attracted much attention since their introduction, in both theory and application aspects. In this paper, we investigate multiple attribute decision-making (MADM) problems with Pythagorean linguistic information based on some new aggregation operators. To begin with, we present some new Pythagorean fuzzy linguistic Muirhead mean (PFLMM) operators to deal with MADM problems with Pythagorean fuzzy linguistic information, including the PFLMM operator, the Pythagorean fuzzy linguistic-weighted Muirhead mean operator, the Pythagorean fuzzy linguistic dual Muirhead mean operator and the Pythagorean fuzzy linguistic dual-weighted Muirhead mean operator. The main advantages of these aggregation operators are that they can capture the interrelationships of multiple attributes among any number of attributes by a parameter vector $P$ and make the information aggregation process more flexible by the parameter vector $P$. In addition, some of the properties of these new aggregation operators are proved and some special cases are discussed where the parameter vector takes some different values. Moreover, we present two new methods to solve MADM problems with Pythagorean fuzzy linguistic information. Finally, an illustrative example is provided to show the feasibility and validity of the new methods, to investigate the influences of parameter vector $P$ on decision-making results, and also to analyze the advantages of the proposed methods by comparing them with the other existing methods.     

Generalized orthopair fuzzy weighted distance-based approximation (WDBA) algorithm in emergency decision-making

With the intensification of global warming trends, the frequent occurrence of natural disasters has brought severe challenges to the sustainable development of society. Emergency decision-making (EDM) in natural disasters is playing an increasingly important role in improving disaster response capacity. In the case of EDM evaluation, the essential problem arises serious incompleteness, impreciseness, subjectivity, and incertitude. The q-rung orthopair fuzzy set (q-ROFS), disposing the indeterminacy portrayed by membership and nonmembership with the sum of $q$th power of them, is a more viable and effective means to seize indeterminacy. The aim of paper is to present a new score function of q-rung orthopair fuzzy number (q-ROFN) for solving the failure problems when comparing two q-ROFNs. Firstly, we introduce some basic set operations for q-ROFS. The properties of these operations are also discussed in detail. Later, we propose a q-rung orthopair fuzzy decision-making method based on weighted distance-based approximation (WDBA), in which the weights of decision-makers are obtained from a nonliner optimization model according to the deviation-based method. Finally, some examples are investigated to illustrate the feasibility and validity of the proposed approach. The salient features of the proposed method, compared to the existing q-rung orthopair fuzzy decision-making methods, are as follows: (a) it can obtain the optimal alternative without counterintuitive phenomena and (b) it has a great power in distinguishing the optimal alternative.     

Efficient secure state estimation against sparse integrity attack for regular linear system

We consider the problem of estimating the state of a time-invariant linear Gaussian system in the presence of integrity attacks. The attacker can compromise $p$ out of $m$ sensors, the set of which is fixed over time and unknown to the system operator, and manipulate the measurements arbitrarily. Under the assumption that the system is regular and system matrix $A$ is non-singular, we propose a secure estimation scheme that is resilient to $p$-sparse attack as long as the system is $2p$-sparse detectable, which achieves the fundamental limit of secure dynamic estimation. In the absence of attack, the proposed estimation coincides with Kalman estimation with a certain probability that can be adjusted to trade-off between performance with and without attack. Furthermore, the detectability condition checking in the designing phase and the estimation computing in the online operating phase are both computationally efficient. Two numerical examples including the IEEE 68 bus test system are provided to corroborate the results and illustrate the performance of the proposed estimator.     

Uncertainty measure based on Tsallis entropy in evidence theory

Dempster-Shafer evidence theory has been widely used in many applications due to its advantages with weaker conditions than Bayes probability. How to measure the uncertainty of basic probability assignment (BPA) in Dempster-Shafer evidence theory is an open and essential issue. Tsallis entropy as nonextensive entropy proposed according to multifractals has been used in many fields. In this paper, a new uncertainty measure of BPA is presented based on Tsallis entropy. The key issue is to determine the value of $q$ in Tsallis entropy. In addition, this paper also analyzes the properties of proposed uncertainty measure. Some numerical examples are used to illustrate the efficiency of the proposed method. Finally, the paper also discusses the application of the proposed method in decision-making.     

An efficient framework for generating robust adversarial examples

Recent studies show that deep neural networks (DNNs) suffer adversarial examples. That is, attackers can mislead the output of a DNN by adding subtle perturbation to a benign input image. In addition, researchers propose new generation of technologies to produce robust adversarial examples. Robust adversarial examples can consistently fool DNN models under predefined hyperparameter space, which can break through some defenses against adversarial examples or even generate physical adversarial examples against real-world applications. Behind these achievements, expectation over transformation (EOT) algorithm plays as the backbone framework for generating robust adversarial examples. Though EOT framework is powerful, we know little about why such a framework can generate robust adversarial examples. To address this issue, we do the first work to explain the principle behind robust adversarial examples. Then, based on the findings, we point out that traditional EOT framework has a performance problem and propose an adaptive sampling algorithm to overcome such a problem. By modeling the sampling process as classic Coupon Collector Problem, we prove that our new framework reduces the cost from $O\text{◂()▸}\left(n\ast \log \mathrm{}\left(n\right)\right)$ to $O\left(n\right)$, where $n$ denotes the number of sampling points. Under the view of computational complexity, the algorithm is optimal for this problem. The experimental results show that our algorithm can save up to 23% overhead in average. This is significant for black-box attack, where the cost is charged by the amount of queries.