Correlation and correlation coefficient of generalized orthopair fuzzy sets

Generalized orthopair fuzzy sets are extensions of ordinary fuzzy sets by relaxing restrictions on the degrees of support for and support against. Correlation analysis is to measure the statistical relationships between two samples or variables. In this paper, we propose a function measuring the interrelation of two -rung orthopair fuzzy sets, whose range is the unit interval . First, the correlation and correlation coefficient of -rung orthopair membership grades are presented, and their basic properties are investigated. Second, these concepts are extended to -rung orthopair fuzzy sets on discrete universes. Then, we discuss their applications in cluster analysis under generalized orthopair fuzzy environments. And, a real-world problem involving the evaluation of companies is used to illustrate the detailed processes of the clustering algorithm. Finally, we introduce the correlation and correlation coefficient of -rung orthopair fuzzy sets on both bounded and unbounded continuous universes and provide some numerical examples to substantiate such arguments.     

TDBF: Two-dimensional belief function

How to efficiently handle uncertain information is still an open issue. In this paper, a new method to deal with uncertain information, named as two-dimensional belief function (TDBF), is presented. A TDBF has two components, T = (), both and are classical belief functions, while is a measure of reliable of . The definition of TDBF and the discounting algorithm are proposed. Compared with the classical discounting model, the proposed TDBF is more flexible and reasonable. Numerical examples are used to show the efficiency and application of the proposed method.     

Research on arithmetic operations over generalized orthopair fuzzy sets

The four fundamental operations of arithmetic for real (and complex) numbers are well known to everybody and quite often used in our daily life. And they have been extended to classical and generalized fuzzy environments with the demand of practical applications. In this paper, we present the arithmetic operations, including addition, subtraction, multiplication, and division operations, over -rung orthopair membership grades, where subtraction and division operations are both defined in two different ways. One is by solving the equation involving addition or multiplication operations, whereas the other is by determining the infimum or supremum of solutions of the corresponding inequality. Not all of -rung orthopairs can be performed by the former method but by the latter method, and it is proved that the former is a special case of the latter. Moreover, the elementary properties of arithmetic operations as well as mixed operations are extensively investigated. Finally, these arithmetic operations are pointwise defined on -rung orthopair fuzzy sets in which the membership degree of each element is a -rung orthopair.     

New logarithmic operational laws and their aggregation operators for Pythagorean fuzzy set and their applications

The aim of this paper is to develop some new logarithm operational laws (LOL) with real number base for the Pythagorean fuzzy sets. Some properties of LOL have been studied and based on these, various weighted averaging and geometric operators have been developed. Then, we utilized it to solve the decision-making problems. The validity of the proposed method is demonstrated with a numerical example and compared the results with the several existing approaches result. Finally, the influences of logarithmic operation and the selection of the logarithmic base in practice are discussed.     

Pythagorean fuzzy Dombi aggregation operators and its applications in multiple attribute decision-making

The operations of -norm and -conorm, developed by Dombi, were generally known as Dombi operations, which may have a better expression of application if they are presented in a new form of flexibility within the general parameter. In this paper, we use Dombi operations to create a few Pythagorean fuzzy Dombi aggregation operators: Pythagorean fuzzy Dombi weighted average operator, Pythagorean fuzzy Dombi order weighted average operator, Pythagorean fuzzy Dombi hybrid weighted average operator, Pythagorean fuzzy Dombi weighted geometric operator, Pythagorean fuzzy Dombi order weighted geometric operator, and Pythagorean fuzzy Dombi hybrid weighted geometric operator. The distinguished feature of these proposed operators is examined. At that point, we have used these operators to build up a model to remedy the multiple attribute decision-making issues under Pythagorean fuzzy environment. Ultimately, a realistic instance is stated to substantiate the created model and to exhibit its applicability and viability.     

Linguistic-induced ordered weighted averaging operator for multiple attribute group decision-making

To solve the problems of making decision with uncertain and imprecise information, Zadeh proposed the concept of Z-number as an ordered pair, the first component of which is a restriction of variable, and the second one is a measure of reliability of the first component. But the decision-makers’ confidence in decision-making was neglected. In this paper, firstly, we present a new method to evaluate and rank -numbers based on the operations of trapezoidal Type 2 fuzzy numbers and generalized trapezoidal fuzzy numbers. Then, linguistic-induced ordered weighted averaging operator and linguistic combined weighted averaging aggregation operator are developed to solve multiple attribute group decision-making problems. And we analyze the main properties of them by utilizing some operational laws of fuzzy linguistic variables. Finally, a numerical example is provided to illustrate the rationality of the proposed method.     

Fermatean fuzzy linguistic set and its application in multicriteria decision making

In this paper, we propose the concept of Fermatean fuzzy linguistic term sets based on linguistic term sets and Fermatean fuzzy sets. The basic operational laws, the score function, and the accuracy function of Fermatean fuzzy linguistic numbers are provided. Then we propose the Fermatean fuzzy linguistic weighted aggregation operator, the Fermatean fuzzy linguistic weighted geometric operator, and the Fermatean fuzzy linguistic distance measures. Furthermore, we extend the TOPSIS method to the proposed distance measures, and the ranking of alternatives is obtained under a Fermatean fuzzy linguistic environment. An example is provided to illustrate the feasibility and effectiveness of the proposed method, and a comparison with the existing method is also analyzed. Finally, the sensitivity analysis of the parameter in the proposed distance measure is also discussed.     

Sparse control for continuous-time systems

In this survey article, we give a comprehensive review of sparse control for continuous-time systems, called maximum hands-off control. The maximum hands-off control is the optimal control, for which we introduce fundamental properties such as necessary conditions, existence, and equivalence to the optimal control. We also show an efficient numerical computation algorithm for the maximum hands-off control based on the time discretization and ADMM (alternating direction method of multipliers). A numerical example is shown with an available MATLAB program.     

Direct synthesis of reduced order H controllers for a class of multivariable plants

Given an nth order, -control input, p-measured output generalized plant, this article proposes a simple, direct approach to design an output feedback H controller with order satisfying for , or for . For this purpose, the output feedback H control problem is transformed into an H state feedback problem for an augmented generalized system. A class of plants for which this transformation always exists and the ensuing controller has order as above, is identified. As a result, for such plants, the reduced order H controller gains are found just by solving a simple linear matrix inequality problem used in state feedback based H control. The efficacy of the proposed approach is studied on some benchmark examples.     

A new -gain analysis framework for discrete-time switched systems based on predictive Lyapunov function

The work proposes the pre--gain analysis framework based on the newly raised nonweighted pre--gain performance index and predictive Lyapunov function, which is devoted to nonweighted -gain analysis and relevant control of discrete-time switched systems under mode-dependent average dwell time. This also provides new ideas for other disturbance-related studies. To begin with, the predictive Lyapunov function is established for switched nonlinear systems in the sense of better reflecting future system dynamics and future external disturbances. Hence, it is achievable to develop less conservative stability and nonweighted pre--gain criteria for switched linear systems. Further, a new disturbance-output expression is devised to match with the nonweighted pre--gain, whose function is to estimate and optimize the traditional nonweighted -gain of the underlying system through discussions. Then, a solvable condition is formulated to seek the piecewise time-dependent gains of switching controller in a convex structure, ensuring the global uniform exponential stability with nonweighted pre--gain and thereby attaining much smaller non-weighted -gain. Finally, the simulation comprised of a circuit system and a numerical example manifests the impressive potential of the obtained results for the purpose of preferable disturbance attenuation performances.     

Mixed coordination mechanisms for scheduling games on hierarchical machines

In this paper, we study scheduling games under mixed coordination mechanisms on hierarchical machines. The two scheduling policies involved are ‐ and ‐, where ‐ (resp., ‐) policy sequences jobs in nondecreasing order of their hierarchies, and jobs of the same hierarchy in nonincreasing (resp., nondecreasing) order of their processing times. We first show the existence of a Nash equilibrium. Then we present the price of anarchy and the price of stability for the games with social costs of minimizing the makespan and maximizing the minimum machine load. All the bounds given in this paper are tight.     

Some Results for Pythagorean Fuzzy Sets

Pythagorean fuzzy sets (PFSs), originally proposed by Yager (Yager, Abbasov. Int J Intell Syst 2013;28:436–452), are a new tool to deal with vagueness considering the membership grades are pairs satisfying the condition . As a generalized set, PFSs have close relationship with intuitionistic fuzzy sets (IFSs). PFSs can be reduced to IFSs satisfying the condition . However, the related operations of PFSs do not take different conditions into consideration. To better understand PFSs, we propose two operations: division and subtraction, and discuss their properties in detail. Then, based on Pythagorean fuzzy aggregation operators, their properties such as boundedness, idempotency, and monotonicity are investigated. Later, we develop a Pythagorean fuzzy superiority and inferiority ranking method to solve uncertainty multiple attribute group decision making problem. Finally, an illustrative example for evaluating the Internet stocks performance is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

PAS: A new powerful and simple quantum computing simulator

In recent years, many researchers have been using CPU for quantum computing simulation. However, in reality, the simulation efficiency of the large-scale simulator is low on a single node. Therefore, striving to improve the simulator efficiency on a single node has become a serious challenge that many researchers need to solve. After many experiments, we found that much computational redundancy and frequent memory access are important factors that hinder the efficient operation of the CPU. This paper proposes a new powerful and simple quantum computing simulator: PAS (power and simple). Compared with existing simulators, PAS introduces four novel optimization methods: efficient hybrid vectorization, fast bitwise operation, memory access filtering, and quantum tracking. In the experiment, we tested the QFT (quantum Fourier transform) and RQC (random quantum circuits) of 21 to 30 qubits and selected the state-of-the-art simulator QuEST (quantum exact simulation toolkit) as the benchmark. After experiments, we have concluded that PAS compared with QuEST can achieve a mean speedup of (QFT), (RQC) (up to , ) on the Intel Xeon E5-2670 v3 CPU.     

Linear-time algorithms for eliminating claws in graphs

Since many -complete graph problems are polynomial-time solvable when restricted to claw-free graphs, we study the problem of determining the distance of a given graph to a claw-free graph, considering vertex elimination a measure. Claw-free Vertex Deletion (CFVD) consists of determining the minimum number of vertices to be removed from a graph such that the resulting graph is claw-free. Although CFVD is -hard in general and recognizing claw-free graphs is still a challenge, where the current best deterministic algorithm for a graph G consists of performing executions of the best algorithm for matrix multiplication, we present linear-time algorithms for CFVD on weighted block graphs and weighted graphs with bounded treewidth. Furthermore, we show that this problem on forests can be solved in linear time by a simpler algorithm, and we determine the exact values for full k-ary trees. On the other hand, we show that CFVD is -hard even when the input graph is a split graph. We also show that the problem is hard to be approximated within any constant factor better than 2, assuming the unique games conjecture.     

Stabilization of the planar vertical take-off and landing using nonlinear feedback control

This article presents a new control strategy for the well-known problem of the planar vertical take-off and landing. The total thrust is computed using a nonlinear feedback compensation so that the altitude reaches the desired altitude. The horizontal position x is then controlled by choosing the orientation angle as a smooth saturation function of x and . A proof of convergence is presented using a Lyapunov approach. The proposed control strategy is successfully tested in numerical simulations.     

Fast vision‐based autonomous detection of moving cooperative target for unmanned aerial vehicle landing

We propose a fast and effective method, fast target detection (FTD), to detect the moving cooperative target for the unmanned aerial vehicle landing, and the target is composed of double circles and a cross. The purpose of our strategy is to land on the target. The FTD method needs to detect the target at the high and low heights. At the high height, the target appears completely and stably in the camera field. The FTD method can detect the circle and cross to rapidly reach the target center, named cross and circle–FTD (). To detect the cross, we propose a slope distance equation to obtain the distance between two slopes. The proposed slopes cluster method, based on the distance equation and K‐means, is used to determine the cross center. At the low height, the target appears incompletely and unstably. Therefore, FTD methods detect only the cross, named cross–FTD (). We extract the cross features ( CFs) based on line segments. Then, four CFs are combined based on graph theory. Experiments on our four datasets show that FTD has rapid speed and good performance. (Our method is implemented in C++ and is available at https://github.com/Li-Zhaoxi/UAV-Vision-Servo .) On the Mohamed Bin Zayed International Robotics Challenge datasets made we constructed, detects the target from a image approximately per pipeline with F‐measure and tracks target approximately per pipeline with F‐measure. detects centers from a image at approximately per image with F‐measure.     

Constrained shortest path tour problem: models,valid inequalities,and Lagrangian heuristics

The constrained shortest path tour problem (CSPTP) is an NP‐hard combinatorial optimization problem defined on a connected directed graph , where V is the set of nodes and A is the set of nonnegative weighted arcs. Given two distinct nodes , an integer value , and node disjoint subsets , , the CSPTP aims at finding the shortest trail from s to t while visiting at least one node in every subset , in this order. In this paper, we perform a comparative analysis between two integer programming (IP) models for the problem. We also propose valid inequalities and a Lagrangian‐based heuristic framework. Branch‐and‐bound algorithms from the literature, as well as a metaheuristic approach, are used for comparison. Extensive computational experiments carried out on benchmark data sets show the effective use of valid inequalities and the quality of bounds obtained by the Lagrangian framework. Because benchmark instances do not require a great computational effort of IP models in the sense that their optimality is reached at the root node of the CPLEX branch‐and‐cut search tree, we introduce new challenging CSPTP instances for which our solution approaches outperform existing ones for the problem.     

Approximate Controllabilty of Semilinear Dynamic Equations on Time Scale

In this paper we study the approximate controllability of semilinear systems on time scale. In order to do so, we first give a complete characterization for the controllability of linear systems on time scale in terms of surjective linear operators in Hilbert spaces. Then we will prove that, under certain conditions on the nonlinear term, if the corresponding linear system is exactly controllable on , for any , then semilinear system on time scale is approximately controllable on .     

A new possibility degree measure for interval‐valued q‐rung orthopair fuzzy sets in decision‐making

As a generalization of the interval‐valued intuitionistic fuzzy sets, a consciousness of interval‐valued $q$‐rung orthopair fuzzy sets (IV $q$‐ROFSs) is a robust and trustworthy tool to fulfill the imprecise information with an adaptation of the manageable parameter $q\ge 1$. However, the ranking of any interval‐valued numbers is very valuable for interval‐valued decision‐making problems. Possibility degree measure is a worthy tool to manage the degree of possibility of one object over the other. Driven by these requisite characteristics, it is fascinating to manifest the possibility degree of comparison between two IV $q$‐ROFSs, and an innovative method is then encouraged to rank the given numbers. Few properties are checked to explain their features and exhibited the advantages of it over the existing possibility measures with some counterintuitive examples. Later on, we consider the multiattribute group decision making (MAGDM) method and embellish it with numerical examples, to rank the alternatives. Several numerical examples are implemented to test the superiority of the stated MAGDM method and to confer its more manageable and adaptable nature.