The practical research on flood risk analysis based on IIOSM and fuzzy α-cut technique

Flood disasters are one of the most common and destructive natural hazards all over the world. In this paper, improved interior-outer-set model (IIOSM) based on information diffusion theory is introduced in detail to assess flood risk in an effort to obtain accurate analytical results that represent the actual situation. Then fuzzy α-cut technique is applied to calculate the fuzzy expected values under the possibility–probability distribution (PPD) calculated by IIOSM. Taking the value of α throughout the interval (0, 1], we correspondingly get access to the conservative risk value (R_C) and venture risk value (R_V). Selection of α, R_C and R_V is dependent on present technical conditions and risk preference of different people. To illustrate the procedure of IIOSM and fuzzy α-cut technique, we employ them respectively to analyze the flood risk in Sanshui District, located in the center of Guangdong province in China. The results, such as risk value estimations, as well as fuzzy expected values, i.e. R_C and R_V under the given α-cut level, can reflect the flood risk quite accurately. The outcomes of this research based on IIOSM and fuzzy α-cut technique offer new insights to carry out an efficient way for various flood protection strategies.     

Fuzzy inference to assess manufacturing process capability with imprecise data

Process capability indices provide numerical measures on whether a process conforms to the defined manufacturing capability prerequisite. These have been successfully applied by companies to compete with and to lead high-profit markets by evaluating the quality and productivity performance. The loss-based process capability index C_pm, sometimes called the Taguchi index, was proposed to measure process capability, wherein the output process measurements are precise. In the present study, we develop a realistic approach that allows the consideration of imprecise output data resulting from the measurements of the products quality. A general method combining the vector of fuzzy numbers to produce the membership function of fuzzy estimator of Taguchi index is introduced for further testing process capability. With the sampling distribution for the precise estimation of C_pm, two useful fuzzy inference criteria, the critical value and the fuzzy P-value, are proposed to assess the manufacturing process capability based on C_pm. The presented methodology takes into the consideration of a certain degree of imprecision on the sample data and leads to the three-decision rule with the four quadrants decision-making plot. The fuzzy inference procedure presented to assess process capability is a natural generalization of the traditional test, when the data are precise the proposed test is reduced to a classical test with a binary decision.     

On distorted probabilities and m-separable fuzzy measures

Fuzzy measures are used in conjunction with fuzzy integrals for aggregation. Their role in the aggregation is to permit the user to express the importance of the information sources (either criteria or experts). Due to the fact that fuzzy measures are set functions, the definition of such measures requires the definition of 2ⁿ parameters, where n is the number of information sources. To make the definition easier, several families of fuzzy measures have been defined in the literature.In this paper m-separable fuzzy measures are introduced. We present some results on this type of measures and we relate them to some of the previous existing ones. We study generating functions for m-separable fuzzy measures and some properties related to these generating functions.     

A fuzzy algorithm for continuous capacitated location allocation model with risk consideration

This paper presents a continuous capacitated location-allocation model with fixed cost as a risk management model. In the presented model, the fixed cost consists of production and installation costs. The model considers risk as percent of unsatisfied demands. The fixed cost is assigned to a zone with a predetermined radius from its center. Because of uncertain environment, demand in each zone is investigated as a fuzzy number. The model is solved by a fuzzy algorithm based on α-cut method. After solving the model based on different α-values, the zones with the largest possibilities are determined for locating new facilities and the best locations are calculated based on the obtained possibilities. Then, the model is solved based on different α-values to determine best allocation values. Also, this paper proposes a Cross Entropy (CE) algorithm considering multivariate normal and multinomial density functions for solving large scale instances and is compared with GAMS. Finally, a numerical example is expressed to illustrate the proposed model.     

Applying fuzzy GERT with approximate fuzzy arithmetic based on the weakest t-norm operations to evaluate repairable reliability

In general, the fuzzy Graphical Evaluation and Review Technique (GERT) usually evaluates/analyzes variables with interval arithmetic (α-cut arithmetic) operations, especially those with complicated fuzzy systems. Thus the interval arithmetic operations may occur accumulating phenomenon of fuzziness in complicated systems, and the accumulating phenomenon of fuzziness may make decision-maker that cannot effectively evaluate problems/systems under vague environment. In order to overcome the accumulating phenomenon of fuzziness or credibly reduce fuzzy spreads, this study adopts approximate fuzzy arithmetic operations under the weakest t-norm arithmetic operations (T_ω) to evaluate fuzzy reliability models based on fuzzy GERT simulation technology. The approximate fuzzy arithmetic operations employ principle of interval arithmetic under the weakest t-norm arithmetic operations. Therefore, the novel fuzzy arithmetic operations may obtain fitter decision values, which have smaller fuzziness accumulating, under vague environment. In numerical examples the approximate fuzzy arithmetic operations has evidenced that it can successfully calculate results of fuzzy operations as interval arithmetic, and can more effectively reduce fuzzy spreads. In the real fuzzy repairable reliability model the performance also shows that the approximate fuzzy arithmetic operations successfully analyze the reliability problem and obtain more confident fuzzy results.     

Hierarchical generation of α-Pareto optimal solutions in large-scale multi-objective non-linear systems with fuzzy parameters

This paper proposes a decomposition method for hierarchical generation of α-Pareto optimal solutions in large-scale multi-objective non-linear programming (MONLP) problems with fuzzy parameters in the objective functions and in the constraints (FMONLP). These fuzzy parameters are characterized by fuzzy numbers. For such problems, the concept of α-Pareto optimality introduced by extending the ordinary Pareto optimality based on the α-level sets of fuzzy numbers. The decomposition method is based on the principle of decompose the original problem into interdependent sub-problems. In this method, the global multi-objective non-linear problem is decomposed into smaller multi-objective sub-problems. The smaller sub-problems, which obtained solved separately by using the weighting method and through an operative procedure. All these solution are coordinates in such a way that an optimal solution for the global problem achieved. In addition, an interactive fuzzy decision-making algorithm for hierarchical generation of α-Pareto optimal solution through the decomposition method is developed. Finally, two numerical examples given to illustrate the results developed in this paper.     

Decision-making in testing process performance with fuzzy data

Over the years, numerous process capability indices (PCIs) have been proposed to the manufacturing industry to provide numerical measures of process performance. Most research efforts have focused on developing and investigating PCIs that assess process capability by precise measurements of output quality. However, real observations of continuous quantities are not precise numbers; in practice, they are more or less imprecise. Since observations of continuous random variables are imprecise the values of related test statistics become imprecise. Therefore, decision rules for statistical tests have to be adapted to this situation. This article presents a set of confidence intervals that produces triangular fuzzy numbers for the estimation of C_pk index using Buckley’s approach with some modification. Additionally, a three-decision testing rule and step-by-step procedure are developed to assess process performance based on fuzzy critical values and fuzzy p-values. This concept is also illustrated with an example for testing process performance.     

Robust H∞ control for a class of switched nonlinear cascade systems via multiple Lyapunov functions approach

This paper addresses the problem of robust H_∞ control for a class of switched nonlinear cascade systems with parameter uncertainty using the multiple Lyapunov functions (MLFs) approach. Each subsystem under consideration is composed of two cascade-connected parts. The uncertain parameters are assumed to be in a known compact set and are allowed to enter the system nonlinearly. Based on the explicit construction of Lyapunov functions, which avoids solving the Hamilton-Jacobi equations, sufficient conditions for the solvability of the robust H_∞ control problem are presented. As an application, the hybrid robust H_∞ control problem for a class of uncertain non-switched nonlinear cascade systems is solved when no single continuous controller is effective. Finally, a numerical example is provided to demonstrate the feasibility of the proposed method.     

Maximal theorems of Menchoff-Rademacher type in non-commutative Lq-spaces

Estimates for maximal functions provide the fundamental tool for solving problems on pointwise convergence. This applies in particular for the Menchoff-Rademacher theorem on orthogonal series in L₂[0,1] and for results due independently to Bennett and Maurey-Nahoum on unconditionally convergent series in L₁[0,1]. We prove corresponding maximal inequalities in non-commutative L_q-spaces over a semifinite von Neumann algebra. The appropriate formulation for non-commutative maximal functions originates in Pisier's recent work on non-commutative vector valued L_q-spaces.     

Reliability evaluation of multi-state consecutive k-out-of-r-from-n: G system

This paper proposes a model that generalizes the linear consecutive k-out-of-r-from-n: G system to multi-state case. In this model the system consists of n linearly ordered multi-state components. Both the system and its components can have different states: from complete failure up to perfect functioning. The system is in state j or above if and only if at least k_j components out of r consecutive are in state j or above. An algorithm is provided for evaluating reliability of a special case of multi-state consecutive k-out-of-r-from-n: G system. The algorithm is based on the application of the total probability theorem and on the application of a special case taken from the [Jinsheng Huang, Ming J. Zuo, Member IEEE and Yanhong Wu, Generalized multi-state k-out-of-n: G system, IEEE Trans. Reliab. 49(1) (2000) 105–111.]. Also numerical results of the formerly published test examples and new examples are given.     

Parameter-dependent H∞ filter design for LPV systems and an autopilot application

This paper considers the design problem of parameter dependent H_∞ filters for linear parameter varying (LPV) systems whose parameters are measurable. Conditions for existence of parameter-dependent Lyapunov function are proposed via parametrical linear matrix inequality (LMI) constraints. Based on the solutions to the LMIs, an algorithm for the gain matrices of LPV filter is presented. The design method is applied to a missile system to demonstrate the effectiveness.     

Some observations on maximum weight stable sets in certain P5-free graphs

The maximum weight stable set problem (MWS) is the weighted version of the maximum stable set problem (MS), which is NP-hard. The class of P₅-free graphs – i.e., graphs with no induced path of five vertices – is the unique minimal class, defined by forbidding a single connected subgraph, for which the computational complexity of MS is an open question. At the same time, it is known that MS can be efficiently solved for (_P5,F)$(P_5\text{,}F)$-free graphs, where F is any graph of five vertices different to a C₅. In this paper we introduce some observations on P₅-free graphs, and apply them to introduce certain subclasses of such graphs for which one can efficiently solve MWS. That extends or improves some known results, and implies – together with other known results – that MWS can be efficiently solved for (_P5,F)$(P_5\text{,}F)$-free graphs where F is any graph of five vertices different to a C₅.     

A note on generalized q-difference equations for q-beta and Andrews–Askey integral

Two q-difference equations with solutions expressed by q-exponential operator identities are investigated. As applications, two extensions of Ramanujan?s formulas for q-beta integral are given, two generalizations of Andrews–Askey integral are obtained. In addition, generating functions for generalized Al-Salam–Carlitz polynomials are deduced. At last, a generalized transformation identity is gained.     

Marginal analysis for the fuzzy p-median problem

The solutions to the fuzzy p-median problem make it possible to leave part of the demand uncovered in order to obtain significant reductions in costs. Moreover, the fuzzy formulation provides the decision-maker with many flexible solutions that he or she may prefer to the classical crisp solution. We introduce some marginal analysis techniques to study how solutions depend on membership functions. Taking into account the internal structure of the problem, we propose a practical criterion to fix the tolerances for the uncovered demand, which happens to be the most sensitive aspect of the fuzzy p-median.     

A new exact exponential penalty function method and nonconvex mathematical programming

A new exact penalty function method, called the l₁ exact exponential penalty function method, is introduced. In this approach, the so-called the exponential penalized optimization problem with the l₁ exact exponential penalty function is associated with the original optimization problem with both inequality and equality constraints. The l₁ exact exponential penalty function method is used to solve nonconvex mathematical programming problems with r-invex functions (with respect to the same function η). The equivalence between sets of optimal solutions of the original mathematical programming problem and of its associated exponential penalized optimization problem is established under suitable r-invexity assumption. Also lower bounds on the penalty parameter are given, for which above these values, this result is true.     

On relative ?0-stability of impulsive hybrid systems via perturbing Lyapunov functions

This paper obtains some sufficient conditions for the relative ?₀-stability of the two impulsive hybrid systems by using the method of perturbing Lyapunov functions.     

Uniqueness of Rankin–Selberg products

In the present paper, we show the equality of the γ  -factors defined by Jacquet, Piatetski-Shapiro and Shalika with those obtained via the Langlands–Shahidi method. Our results are new in the case of positive characteristic, where we establish a refined version of the local–global principle for _GLn${\mathrm{GL}}_n$ which has independent interest. In characteristic zero, the results are due to Shahidi. The comparison of γ-factors is made via a uniqueness result for Rankin–Selberg γ-factors.     

Bose–Einstein condensation in satisfiability problems

This paper is concerned with the complex behavior arising in satisfiability problems. We present a new statistical physics-based characterization of the satisfiability problem. Specifically, we design an algorithm that is able to produce graphs starting from a k-SAT instance, in order to analyze them and show whether a Bose–Einstein condensation occurs. We observe that, analogously to complex networks, the networks of k-SAT instances follow Bose statistics and can undergo Bose–Einstein condensation. In particular, k-SAT instances move from a fit-get-rich network to a winner-takes-all network as the ratio of clauses to variables decreases, and the phase transition of k-SAT approximates the critical temperature for the Bose–Einstein condensation. Finally, we employ the fitness-based classification to enhance SAT solvers (e.g., ChainSAT) and obtain the consistently highest performing SAT solver for CNF formulas, and therefore a new class of efficient hardware and software verification tools.     

H∞ performance for uncertain discrete switched systems with interval time-varying delay via switching signal design

The switching signal design for H_∞ performance of uncertain discrete switched systems with interval delay and linear fractional perturbations is considered in this paper. Some LMI stability criteria are proposed to design the switching signal and guarantee the H_∞ performance for discrete switched time-delay system. Some nonnegative inequalities are introduced to improve the conservativeness of the proposed results. A numerical example is illustrated to show the less conservativeness of the main result. Finally, a water quality model is also provided to demonstrate the practical applications of our proposed results.     

Nonlinear programming with E-preinvex and local E-preinvex functions

In this paper, we extend the class of E-convex sets, E-convex and E-quasiconvex functions introduced by [Youness, E.A., 1999. E-convex sets, E-convex functions and E-convex programming. Journal of Optimization Theory and Applications 102, 439–450], respectively by [Syau, Yu-Ru, Lee, E. Stanley, 2005. Some properties of E-convex functions. Applied Mathematics Letters 18, 1074–1080] to E-invex set, E-preinvex, E-prequasiinvex and corresponding local concepts. Some properties of these classes are studied. As an application of our results, we consider the nonlinear programming problem for which, we establish that, under mild conditions, a local minimum is a global minimum.