Another view on q‐rung orthopair fuzzy sets

In this paper, two new approaches have been presented to view q‐rung orthopair fuzzy sets. In the first approach, these can viewed as L‐fuzzy sets, whereas the second approach is based on the notion of orbits. Uncertainty index is the quantity , which remains constant for all points in an orbit. Certain operators can be defined in q‐ROF sets, which affect when applied to some q‐ROF sets. Operators , , and have been defined. It is studied that how these operators affect when applied to some q‐ROF set A.     

A General Version of the Triple Π Operator

Recent developments of sensors and computers have raised the problem of handling huge amounts of complex data that users try to synthesize for decision making. Aggregation operators, such as those appearing in fuzzy sets theory, are useful tools for this synthesis but in their present formulation, these operators only deal with a finite set of arguments. In this paper, we introduce , an extension of both Yager–Rybalov Triple Π and Mean Triple Π operators to general measure spaces that can deal with temporal or spatiotemporal intensive data streams. Known properties and inequalities are extended in this more general setting. The notion of moving is also introduced and it can be applied to a solar radiation data stream. This may lead to further works on data fusion and on similar extensions of some other operators.     

Inference in Hybrid Bayesian Networks with Nonlinear Deterministic Conditionals

To enable inference in hybrid Bayesian networks (BNs) containing nonlinear deterministic conditional distributions, Cobb and Shenoy in 2005 propose approximating nonlinear deterministic functions by piecewise linear (PL) ones. In this paper, we describe a method for finding PL approximations of nonlinear functions based on a penalized mean square error (MSE) heuristic, which consists of minimizing a penalized MSE function subject to two principles, domain and symmetry. We illustrate our method for some commonly used one‐dimensional and two‐dimensional nonlinear deterministic functions such as , , , and . Finally, we solve two small examples of hybrid BNs containing nonlinear deterministic conditionals that arise in practice.     

An Axiomatic System for Conditional Attribute Implications in Triadic Concept Analysis

In this paper, we define a sound and complete inference system for triadic implications generated from a formal triadic context , where G, M, and B are object, attribute, and condition sets, respectively, and I is a ternary relation . The inference system is expressed as a set of axioms “à la Armstrong.” The type of triadic implications we are considering in this paper is called conditional attribute implication (CAI) and has the following form: , where X and Y are subsets of M, and is a subset of B. Such implication states that Ximplies Y under all conditions in and any subset of it. Moreover, we propose a method to compute CAIs from Biedermann's implications. We also introduce an algorithm to compute the closure of an attribute set X w.r.t. a set Σ of CAIs given a set of conditions.     

A Note on Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets

In this note, we point out an error to the proof of Theorem 3.4 in Zhang and Xu (Int J Intell Syst 2014;29(12):1061–1078) by a counterexample. We find that the inequality (i.e., ) with respect to the degrees of indeterminacy of any three Pythagorean fuzzy numbers in the proof of Theorem 3.4 in Zhang and Xu's paper is not valid. A new proof is provided in this note.     

Parametric Weighting Function for WOWA Operator and Its Application in Decision Making

We propose a parametric weighting function for weighted ordered weighted averaging (WOWA) operator in decision making under uncertainty and risk environment. The weighting function is not only related to the attitude of the decision makers, but also to the event probability. Under the proposed method, the decision makers underweight small probability gains and overweight large probability gains, which is right in consistence with the psychology of the decision makers in the prospect theory. Meanwhile, it generalizes two specific weighting functions and proposed by Torra (International Journal of Intelligent Systems 12 (1997) 153–166). Moreover, the expected utility becomes a special case of the parametric weighting function for WOWA operator. To illustrate its application, an example of introducing and pricing new product is given, which shows the optimal strategies corresponding to different parameter values r.     

An Efficient Algorithm for Determining an Aesthetic Shape Connecting Unorganized 2D Points

We present anefficient algorithm for determining an aesthetically pleasing shape boundary connecting all the points in a given unorganized set of 2D points, with no other information than point coordinates. By posing shape construction as a minimisation problem which follows the Gestalt laws, our desired shape is non‐intersecting, interpolates all points and minimizes a criterion related to these laws. The basis for our algorithm is an initial graph, an extension of the Euclidean minimum spanning tree but with no leaf nodes, called as the minimum boundary complex . and can be expressed similarly by parametrizing a topological constraint. A close approximation of , termed can be computed fast using a greedy algorithm. is then transformed into a closed interpolating boundary in two steps to satisfy ’s topological and minimization requirements. Computing exactly is an NP (Non‐Polynomial)‐hard problem, whereas is computed in linearithmic time. We present many examples showing considerable improvement over previous techniques, especially for shapes with sharp corners. Source code is available online.     

Full complexity analysis of the diameter‐constrained reliability

Let be a simple graph with nodes and links, a subset of “terminals,” a vector , and a positive integer d, called “diameter.” We assume that nodes are perfect but links fail stochastically and independently, with probabilities . The “diameter‐constrained reliability” (DCR) is the probability that the terminals of the resulting subgraph remain connected by paths composed of d links, or less. This number is denoted by . The general DCR computation belongs to the class of ‐hard problems, since it subsumes the problem of computing the probability that a random graph is connected. The contributions of this paper are twofold. First, a full analysis of the computational complexity of DCR subproblems is presented in terms of the number of terminal nodes and the diameter d. Second, we extend the class of graphs that accept efficient DCR computation. In this class, we include graphs with bounded co‐rank, graphs with bounded genus, planar graphs, and, in particular, Monma graphs, which are relevant to robust network design.     

Determinization and minimization of finite acyclic automata by incremental techniques

The determinization of a nondeterministic finite automaton (FA) is the process of generating a deterministic FA (DFA) equivalent to (sharing the same regular language of) . The minimization of is the process of generating the minimal DFA equivalent to . Classical algorithms for determinization and minimization are available in the literature for several decades. However, they operate monolithically, assuming that the FA to be either determinized or minimized is given once and for all. By contrast, we consider determinization and minimization in a dynamic context, where augments over time: after each augmentation, determinization and minimization of into is required. Using classical monolithic algorithms to solve this problem is bound to poor performance. An algorithm for incremental determinization and minimization of acyclic finite automata, called IDMA, is proposed. Despite being conceived within the narrow domain of model‐based diagnosis and monitoring of active systems, the algorithm is general‐purpose in nature. Experimental evidence indicates that IDMA is far more efficient than classical algorithms in solving incremental determinization and minimization problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Application of generally weighted moving average method to tracking signal state space model

In predicting time series, if a trend includes a structural break, then a state space model can be applied to revise the predictive method. Some scholars suggest that restricted damped trend models yield excellent prediction results by automatically revising unforeseen structural break factors in the prediction process. Restricted damped trend models add a smoothed error statistic to a local‐level model and use the exponentially weighted moving average (EWMA) method to make corrections. This paper applies the generally weighted moving average (GWMA) concept and method to a restricted damped trend model that changes the smoothed error statistic from the EWMA form to the GWMA form and adds the correction parameter λ, which distinguishes three situations , , and . The original restricted damped trend model applies only to , enabling the model to capture situations in which and increases its generality. This paper also compares the effect of various parameter values on the predictive model and finds the range of parameter settings that optimize the model.     

Enhancing Bayesian Estimators for Removing Camera Shake

The aim of removing camera shake is to estimate a sharp version x from a shaken image y when the blur kernel k is unknown. Recent research on this topic evolved through two paradigms called and . only solves for k by marginalizing the image prior, while recovers both x and k by selecting the mode of the posterior distribution. This paper first systematically analyses the latent limitations of these two estimators through Bayesian analysis. We explain the reason why it is so difficult for image statistics to solve the previously reported failure. Then we show that the leading methods, which depend on efficient prediction of large step edges, are not robust to natural images due to the diversity of edges. , although much more robust to diverse edges, is constrained by two factors: the prior variation over different images, and the ratio between image size and kernel size. To overcome these limitations, we introduce an inter‐scale prior prediction scheme and a principled mechanism for integrating the sharpening filter into . Both qualitative results and extensive quantitative comparisons demonstrate that our algorithm outperforms state‐of‐the‐art methods.     

Mixed control of hidden Markov jump systems

In this work, we study the mixed control for Markov jump linear systems with hidden Markov parameters. The hidden Markov process is denoted by , where the nonobservable component θ(k) represents the mode of operation of the system, whereas represents the observable component provided by a detector. The goal is to obtain design techniques for mixed control problems, with the controllers depending only on the estimate , for problems formulated in 3 different forms: (i) minimizing an upper bound on the norm subject to a given restriction on the norm; (ii) minimizing an upper bound on the norm, while limiting the norm; and (iii) minimizing a weighted combination of upper bounds of both the and norms. We propose also new conditions for synthesizing robust controllers under parametric uncertainty in the detector probabilities and in the transition probabilities. The so‐called cluster case for the mixed control problem is also analyzed under the detector approach. The results are illustrated by means of 2 numerical examples.     

Design and analysis of wideband U‐slot patch antenna with U‐shaped parasitic elements

A single layer single probe‐fed wideband microstrip antenna is presented and investigated. By cutting a U‐slot in the rectangular patch, and by incorporating two identical U‐shaped parasitic patches around both the radiating edges and the nonradiating edges of the rectangular patch, three resonant frequencies are excited to form the wideband performance. Details of the antenna design is presented. The measured and simulated results are in good agreement, the measured impedance bandwidth is GHz ( GHz), or centered at GHz, which covers WLAN GHz ( GHz), WLAN GHz ( GHz), and WIMAX GHz ( GHz) bands. The measured peak gains at the three resonant frequencies are dB, dB, and dB, respectively. An equivalent circuit model which is based on the transmission line theory, the asymmetric coupled microstrip lines theory, and the π‐network theory is established. This equivalent circuit model is used to give an insight into the wideband mechanism of the proposed antenna, and is also used to explain why the three resonant frequencies shift at the variations of different parameters from a physical point of view. The error analysis is given to demonstrate the validity of the equivalent circuit model.     

Polynomial time approximation algorithms for proportionate open‐shop scheduling

This paper deals with the presentation of polynomial time (approximation) algorithms for a variant of open‐shop scheduling, where the processing times are only machine‐dependent. This variant of scheduling is called proportionate scheduling and its applications are used in many real‐world environments. This paper develops three polynomial time algorithms for the problem. First, we present a polynomial time algorithm that solves the problem optimally if , where n and m denote the numbers of jobs and machines, respectively. If, on the other hand, , we develop a polynomial time approximation algorithm with a worst‐case performance ratio of that improves the bound existing for general open‐shops. Next, in the case of , we take into account the problem under consideration as a master problem and convert it into a simpler secondary approximation problem. Furthermore, we formulate both the master and secondary problems, and compare their complexity sizes. We finally present another polynomial time algorithm that provides optimal solution for a special case of the problem where .     

On computing the 2‐diameter ‐constrained K ‐reliability of networks

This article considers a communication network modeled by a graph and a distinguished set of terminal nodes . We assume that the nodes never fail, but the edges fail randomly and independently with known probabilities. The classical K ‐reliability problem computes the probability that the subnetwork is composed only by the surviving edges in such a way that all terminals communicate with each other. The d ‐diameter ‐constrained K ‐reliability generalization also imposes the constraint that each pair of terminals must be the extremes of a surviving path of approximately d length. It allows modeling communication network situations in which limits exist on the acceptable delay times or on the amount of hops that packets can undergo. Both problems have been shown to be NP ‐hard, yet the complexity of certain subproblems remains undetermined. In particular, when , it was an open question whether the instances with were solvable in polynomial time. In this paper, we prove that when and is a fixed parameter (i.e. not an input) the problem turns out to be polynomial in the number of nodes of the network (in fact linear). We also introduce an algorithm to compute these cases in such time and also provide two numerical examples.     

Graphical tuning method of FOPID controllers for fractional order uncertain system achieving robust ‐stability

This paper focuses on the graphical tuning method of fractional order proportional integral derivative (FOPID) controllers for fractional order uncertain system achieving robust ‐stability. Firstly, general result is presented to check the robust ‐stability of the linear fractional order interval polynomial. Then some alternative algorithms and results are proposed to reduce the computational effort of the general result. Secondly, a general graphical tuning method together with some computational efficient algorithms are proposed to determine the complete set of FOPID controllers that provides ‐stability for interval fractional order plant. These methods will combine the results for fractional order parametric robust control with the method of FOPID ‐stabilization for a fixed plant. At last, two important extensions will be given to the proposed graphical tuning methods: determine the ‐stabilizing region for fractional order systems with two kinds of more general and complex uncertainty structures: multi‐linear interval uncertainty and mixed‐type uncertainties. Numerical examples are followed to illustrate the effectiveness of the method. Copyright © 2015 John Wiley & Sons, Ltd.     

Computational and structural analysis of the contour of graphs

Let be a finite, simple, and connected graph. The closed interval of a set is the set of all vertices lying on a shortest path between any pair of vertices of S. The set S is geodetic if . The eccentricity of a vertex v is the number of edges in the greatest shortest path between v and any vertex w of G. A vertex v is a contour vertex if no neighbor of v has eccentricity greater than v. The contour of G is the set formed by the contour vertices of G. We consider two problems: the problem of determining whether the contour of a graph class is geodetic; the problem of determining if there exists a graph such that is not geodetic. We obtain a sufficient condition that is useful for both problems; we prove a realization theorem related to problem and show two infinite families such that is not geodetic. Using computational tools, we establish the minimum graphs for which is not geodetic; and show that all graphs with , and all bipartite graphs with , are such that is geodetic.     

Greedy Geometric Algorithms for Collection of Balls,with Applications to Geometric Approximation and Molecular Coarse‐Graining

Choosing balls that best approximate a 3D object is a non‐trivial problem. To answer it, we first address the inner approximation problem, which consists of approximating an object defined by a union of n balls with balls defining a region . This solution is further used to construct an outer approximation enclosing the initial shape, and an interpolated approximation sandwiched between the inner and outer approximations. The inner approximation problem is reduced to a geometric generalization of weighted max k‐cover, solved with the greedy strategy which achieves the classical lower bound. The outer approximation is reduced to exploiting the partition of the boundary of by the Apollonius Voronoi diagram of the balls defining the inner approximation. Implementation‐wise, we present robust software incorporating the calculation of the exact Delaunay triangulation of points with degree two algebraic coordinates, of the exact medial axis of a union of balls, and of a certified estimate of the volume of a union of balls. Application‐wise, we exhibit accurate coarse‐grain molecular models using a number of balls 20 times smaller than the number of atoms, a key requirement to simulate crowded cellular environments.     

Thin‐slot formalism for the split‐field FDTD analysis of periodic structure

An expression of the thin‐slot formalism is presented to alleviate the gridding of the split‐field finite‐difference time‐domain (FDTD) solution for periodic structure. The varying auxiliary‐field ( , ) and split‐field ( , ) distributions near the slots are analytically derived from the varying field ( , ). The update equations for the split‐field FDTD are obtained by incorporating those varying field distributions into the split‐field equations in integral form. A frequency selective surface (FSS) structure is applied to verify the proposed method. The results indicate that the computational efficiency is improved.     

Low Frequency Sensitivity Function Constraints for Nonlinear ‐stable Networked Control

The paper derives a robust networked controller design method for systems with saturation where the delay is large and uncertain, as in one‐directional data flow‐control. A classical linear H_∞ criterion is first formulated in terms of the sensitivity and complementary sensitivity functions. A new asymptotic constraint is then derived, which specifies the minimum amount of low frequency gain that is needed in the sensitivity function to conclude on non‐linear closed loop ‐stability using the Popov criterion. This result guides the selection of the design criterion, thereby adjusting the linear controller design for better handling of delay and saturation. The controller design method then uses gridding to pre‐compute a subset of the stability region. Based on the pre‐computed region, a robust ‐stable controller can be selected. Alternatively, an adaptive controller could recompute ‐stable controllers on‐line using the pre‐computed region. Simulations show that the controller meets the specified stability and performance requirements.