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The framework of interval-valued fuzzy preference relations (IVFPRs) is adequate and effective to model human preference evaluations under indeterminacy. This paper analyzes three recently presented multiplicative transitivity models of IVFPRs and exposes their drawbacks. An and-like-uninorm-based functional transitivity equation is developed to introduce a multiplicative consistency notion for IVFPRs. Based on the transitivity logarithmic equation, a geometric-consistency index is further proposed to compute the inconsistency level of an IVFPR. The paper builds a logarithmic least squares model with row indeterminacy constraints and equivalently transforms it into a quadratic programming model for finding a closed-form solution of the normalized interval-valued fuzzy weights of IVFPRs. A novel method is subsequently presented to check the acceptability of an IVFPR by examining its acceptable consistency and acceptable indeterminacy. An approach including an and-like-uninorm-based maximization model is introduced to aggregate local interval-valued fuzzy weights and an interval fuzzy analytic hierarchy process is designed step-by-step. An illustrative example and a comparison study are utilized to demonstrate the performance and merits of the presented models. Meanwhile, an outstanding undergraduate student selection problem in international exchange is provided to show the application of the proposed decision method.

In this paper, we derive conditions for the existence of Pareto efficient strategy and Pareto solution under H∞ constraint for the linear quadratic (LQ) finite horizon cooperative differential game of stochastic systems with state-, control- and disturbance-multiplicative noise. Firstly, we present a stochastic bounded real lemma (SBRL) with any initial condition for the considered stochastic system. Next, a necessary and sufficient condition for Pareto optimal strategy under the H∞ constraint is researched by two cross-coupled generalized differential Riccati equations (GDREs), which is the key contribution of this paper. Another contribution is that the Pareto solution for every controller is given under the Pareto efficient strategy and worst-case external disturbance . Finally, a practical example is given to illustrate the effectiveness of our results.

In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated In doing so, the feasible region is formed by the intersection of two inequality fuzzy systems and Dubois-Prade family of t-norms which are considered as fuzzy composition. The most well-known continuous t-norms are Archimedean such as Frank, Yager, Hamacher, Sugeno-Weber and Schweizer-Sklar family. An interesting family of t-norms that is not Archimedean has been introduced by Dubois and Prade. In this paper, the resolution of the feasible region of the problem is initialy investigated when it is defined with max-Duboise-Prade composition. A necessary and sufficient condition along with three other necessary conditions are derived for determining the feasibility. of the problem. Moreover, two procedures have also been presented with the aim of simplifying the current linear problems. A method is proposed to generate random feasible max-Dubois-Prade fuzzy relational inequalities and an algorithm is accordingly presented to solve the problem. Finally, an example is described to illustrate this algorithm.

In this paper, we propose a new autocratic multiattribute group decision making (AMAGDM) method for hotel location selection based on interval-valued intuitionistic fuzzy sets (IVIFSs), where the evaluating values of the attributes for alternatives and the weights of the attributes given by decision makers are represented by interval-valued intuitionistic fuzzy values (IVIFVs). The proposed method calculates the changing of the weights of the decision makers until the group consensus degree (GCD) of the decision makers is larger than or equal to a predefined threshold value. We also apply the proposed AMAGDM method to deal with the hotel location selection problem. The main contribution of this paper is that we propose a new AMAGDM method which is simpler than Wibowo's method (2013), where the drawback of Wibowo's method is that it is too complicated due to the fact that it adopts the concept of ideal solutions for determining the overall performance of each hotel location alternative with respect to all the selection criteria. The proposed AMAGDM method provides us with a very useful way for AMAGDM in interval-valued intuitionistic fuzzy environments.

In this paper, optimization of a linear objective function subject to a generalized fuzzy relational inequalities is investigated in which an arbitrary continuous t-norm is considered as fuzzy composition, and fuzzy inequality replaces ordinary inequality in the constraints. Unlike most optimization algorithms, in fuzzy relational inequalities with fuzzy constraints (FRI-FC) we find a near-feasible solution having a better objective value than those resulting from the resolution of the similar problems with crisp (ordinary) inequality constraints. Such solutions are called super-optima in this paper. Subsequently, an algorithm is proposed to find a super-optimum with pre-specified desirable infeasibility. For this purpose, we firstly study some structural properties of the FRI-FC problem and present a new formulization that is independent of t-norms used in the constraints of the problem. This new formulation converts the primary problem into an equivalent problem with simple constraints without considering any penalty parameters. However, it is proved that the transformed equivalent problem enables our algorithm to distinguish unfavorable points and generate a sequence of solutions converging to a super-optimum under a certain sufficient condition. Finally, a modified PSO is presented in which the ability of PSO to solve unconstrained problems with continuous domain, the structure of the transformed problem and some fuzzy structural modifications are combined and form an efficient algorithm to solve the generalized FRI-FC problems. The modified PSO algorithm has been applied to the generalized FRI-FC problem defined with ten well-known continuous t-norms. Additionally, an idea of the FRI-FC problems as outer approximators for FRI problems with ordinary inequalities is also investigated.

The paper deals with two crucial steps in multi-criteria decision analysis, that are consistency of the judgments and priority vectors for alternatives/criteria. From a side, several consistency indices are proposed for measuring the consistency of a Pairwise Comparison Matrix. From another one, conditions weaker than consistency, such as transitivity and weak consistency, are proposed for representing further levels of coherence of a Decision Maker when he/she expresses his/her preferences by means of reciprocal Pairwise Comparison Matrix. Firstly, in this paper, a simulation is performed in order to establish a relation between random consistency index and coherence level. Then, since weak consistency ensures reliability to priority vectors proposed in literature, a second simulation is performed in order to measure, in case of no weak consistency, the reliability of these priority vectors.

In this paper, we propose a new multiattribute decision making (MADM) method based on the interval-valued intuitionistic fuzzy weighted geometric average (IIFWGA) operator, the accuracy function of interval-valued intuitionistic fuzzy values (IVIFVs) and particle swarm optimization (PSO) techniques, where the weights of attributes and the evaluating values of alternatives with respect to attributes are represented by IVIFVs. First, the proposed method uses an accuracy function to transform the decision matrix given by the decision maker and represented by IVIFVs into a transformed decision matrix represented by real values in [1,1]. Then, it produces the optimal weights of the attributes based on the obtained transformed decision matrix and PSO techniques. It determines the weighted evaluating IVIFV of each alternative based on the IIFWGA operator, the obtained optimal weights of the attributes and the decision matrix given by the decision maker represented by IVIFVs. Finally, it calculates the transformed value of the weighted evaluating IVIFV of each alternative based on the accuracy function to get the preference order of the alternatives. The main contribution of this paper is that we propose a new MADM method based on the IIFWGA operator of IVIFVs, the accuracy function of IVIFVs and PSO techniques, which can overcome the drawbacks of the existing MADM methods for MADM in interval-valued intuitionistic fuzzy (IVIF) environments.

Over the last two decades numerous metaheuristics have been proposed and it seems today that nobody is able to understand, evaluate, or compare them all. In principle, optimization methods, including the recently popular Evolutionary Computation or Swarm Intelligence-based ones, should be developed in order to solve real-world problems. Yet the vast majority of metaheuristics are tested in the source papers on artificial benchmarks only, so their usefulness for various practical applications remains unverified. As a result, choosing the proper method for a particular real-world problem is a difficult task. This paper shows that such a choice is even more complicated if one wishes, with good reason, to use metaheuristics twice, once to find the best and then to find the worst solutions for the specific numerical real-world problem. It often occurs that for either case different optimizers are to be recommended. The above finding is based on testing 30 metaheuristics on numerical real-world problems from CEC2011. First we solve 22 minimization problems as defined for CEC2011. Then we reverse the objective function for each problem and search for its maximizing solution. We also observe that algorithms that are highly ranked on average may not perform best for any given specific problem. Rather, the highest average ranking may be achieved by methods that are never among the poorest ones. In other words, occasional winners may get less attention than rare losers.

Many-objective problems refer to the optimization problems containing more than three conflicting objectives. To obtain a representative set of well-distributed non-dominated solutions close to Pareto front in the objective space remains a challenging problem. Many papers have proposed different Multi-Objective Evolutionary Algorithms to solve the lack of the convergence and diversity in many-objective problems. One of the more promising approaches uses a set of reference points to discriminate the solutions and guide the search process. However, this approach was incorporated mainly in Multi-Objective Evolutionary Algorithms, and there are just some few promising adaptations of Particle Swarm Optimization approaches for effectively tackling many-objective problems regarding convergence and diversity. Thus, this paper proposes a practical and efficient Many-Objective Particle Swarm Optimization algorithm for solving many-objective problems. Our proposal uses a set of reference points dynamically determined according to the search process, allowing the algorithm to converge to the Pareto front, but maintaining the diversity of the Pareto front. Our experimental results demonstrate superior or similar performance when compared to other state-of-art algorithms.

Estimation of Distribution Algorithms (EDAs) constitutes a class of evolutionary algorithms that can extract and exploit knowledge acquired throughout the optimization process. The most critical step in the EDAs is the estimation of the joint probability distribution associated to the variables from the most promising solutions determined by the evaluation function. Recently, a new approach to EDAs has been developed, based on copula theory, to improve the estimation of the joint probability distribution function. However, most copula-based EDAs still present two major drawbacks: focus on copulas with constant parameters, and premature convergence. This paper presents a new copula-based estimation of distribution algorithm for numerical optimization problems, named EDA based on Multivariate Elliptical Copulas (EDA-MEC). This model uses multivariate copulas to estimate the probability distribution for generating a population of individuals. The EDA-MEC differs from other copula-based EDAs in several aspects: the copula parameter is dynamically estimated, using dependence measures; it uses a variation of the learned probability distribution to generate individuals that help to avoid premature convergence; and uses a heuristic to reinitialize the population as an additional technique to preserve the diversity of solutions. The paper shows, by means of a set of parametric tests, that this approach improves the overall performance of the optimization process when compared with other copula-based EDAs and with other efficient heuristics such as the Covariance Matrix Adaptation Evolution Strategy (CMA-ES).

peer-reviewed The minimum cross-entropy principle is an established technique for design of an unknown distribution, processing linear functional constraints on the distribution. More generally, fully probabilistic design (FPD) chooses the distribution—within the knowledge-constrained set of possible distributions—for which the Kullback-Leibler divergence to the designer’s ideal distribution is minimized. These principles treat the unknown distribution deterministically. In this paper, fully probabilistic design is applied to hierarchical Bayesian models for the first time, yielding optimal design of a (possibly nonparametric) stochastic model for the unknown distribution. This equips minimum cross-entropy and FPD distributional estimates with measures of uncertainty. It enables robust choice of the optimal model, as well as randomization of this choice. The ability to process non-linear functional constraints in the constructed distribution significantly extends the applicability of these principles. Currently available FPD procedures for (a) merging of external knowledge, (b) approximate learning and stabilized forgetting, (c) decision strategy design, and (d) local adaptive control design, are unified for the first time via the hierarchical FPD framework of this paper.

This Paper Puts Forward A Comprehensive Study Of The Design Of Global Stopping Criteria For Multi-Objective Optimization. In This Study We Propose A Global Stopping Criterion, Which Is Terms As Mgbm After The Authors Surnames. Mgbm Combines A Novel Progress Indicator, Called Mutual Domination Rate (Mdr) Indicator, With A Simplified Kalman Filter, Which Is Used For Evidence-Gathering Purposes. The Mdr Indicator, Which Is Also Introduced, Is A Special-Purpose Progress Indicator Designed For The Purpose Of Stopping A Multi-Objective Optimization. As Part Of The Paper We Describe The Criterion From A Theoretical Perspective And Examine Its Performance On A Number Of Test Problems. We Also Compare This Method With Similar Approaches To The Issue. The Results Of These Experiments Suggest That Mgbm Is A Valid And Accurate Approach. (C) 2016 Elsevier Inc. All Rights Reserved. This work was funded in part by CNPq BJT Project 407851/2012-7 and CNPq PVE Project 314017/2013-5

Propose the improved augmented Lagrangian method (iALF) to deal with constraints in a more efficient way.Propose cooperative coevolutionary method using improved augmented Lagrangian (CCiALF) approach for solving constrained optimisation problems.Solve the CEC'2006 constrained optimisation test suite with proposed CCiALF algorithm with higher solution quality compared to the state-of-the-art algorithms.Solve practical engineering problems by proposed CCiALF algorithm and improve the best known objective function value that exists in the literature. In constrained optimisation, the augmented Lagrangian method is considered as one of the most effective and efficient methods. This paper studies the behaviour of augmented Lagrangian function (ALF) in the solution space and then proposes an improved augmented Lagrangian method. We have shown that our proposed method can overcome some of the drawbacks of the conventional augmented Lagrangian method. With the improved augmented Lagrangian approach, this paper then proposes a cooperative coevolutionary differential evolution algorithm for solving constrained optimisation problems. The proposed algorithm is evaluated on a set of 24 well-known benchmark functions and five practical engineering problems. Experimental results demonstrate that the proposed algorithm outperforms the state-of-the-art algorithms with respect to solution quality as well as efficiency.

In this paper, the robust optimal control of continuous-time affine nonlinear systems with matched uncertainties is investigated by using a data-based integral policy iteration approach. It is a natural extension of the traditional optimal control design, under the framework of adaptive dynamic programming (ADP) method, to robust optimal control of nonlinear systems with matched uncertainties. In theoretical aspect, by increasing a feedback gain to the optimal controller of the nominal system, the robust controller of the matched uncertain system is obtained, which also achieves optimality with a newly well-defined cost function. When regarding the implementation, the data-based integral policy iteration algorithm is used to solve the Hamilton-Jacobi-Bellman equation corresponding to the nominal system with completely unknown dynamics information. Then, the actor-critic technique based on neural networks and least squares implementation method are employed to facilitate deriving the optimal control law iteratively, so that the closed-form expression of the robust optimal controller is available. Additionally, two simulation examples with application backgrounds are presented to illustrate the effectiveness of the established robust optimal control scheme. In summary, it is important to note that the result developed in this paper broadens the application scope of ADP-based optimal control approach to more general nonlinear systems possessing dynamical uncertainties.

This paper considers parameter identification problems for multivariable Hammerstein controlled autoregressive moving average (CARMA) systems. We transform a multivariable Hammerstein CARMA system into a new identification model which contains a set of bilinear parameter vectors, i.e., two sets of parameter vectors. To solve the difficulty of estimating the two sets of parameter vectors, this paper constructs two different forms of the identification model in each of which the output is linear about one set of parameter vectors, and presents an extended stochastic gradient algorithm to interactively estimate the two sets of parameter vectors by using the hierarchical identification principle. The proposed hierarchical identification algorithm has a higher computational efficiency than the over-parametrization model based stochastic gradient algorithm. The simulation results indicate that the proposed algorithm is effective, and an example of modeling a photovoltaic array system with the multivariable Hammerstein model is given.

This paper proposes an enhanced Particle Swarm Optimisation (PSO) algorithm and examines its performance. In the proposed PSO approach, PSO is combined with Evolutionary Game Theory to improve convergence. One of the main challenges of such stochastic optimisation algorithms is the difficulty in the theoretical analysis of the convergence and performance. Therefore, this paper analytically investigates the convergence and performance of the proposed PSO algorithm. The analysis results show that convergence speed of the proposed PSO is superior to that of the Standard PSO approach. This paper also develops another algorithm combining the proposed PSO with the Standard PSO algorithm to mitigate the potential premature convergence issue in the proposed PSO algorithm. The combined approach consists of two types of particles, one follows Standard PSO and the other follows the proposed PSO. This enables exploitation of both diversification of the particles’ exploration and adaptation of the search direction.

This paper investigated the multiple performance constrained sliding mode fuzzy control problem for continuous-time stochastic nonlinear systems with multiplicative noises. The stochastic nonlinear systems considered in this paper are represented by Takagi-Sugeno (T-S) fuzzy models with a bilinear consequent part. To simultaneously satisfy stability, passivity and variance constraints, sufficient conditions are derived in terms of linear matrix inequalities based on Lyapunov theory. By solving the corresponding sufficient conditions, a parallel distributed compensation based fuzzy controller is obtained to ensure the stability of the closed-loop nonlinear systems subject to variance and passivity performance constraints. The main contribution of this paper is a sliding mode fuzzy controller developed such that the considered multiple performance constraints can be satisfied for the T-S fuzzy systems with a bilinear consequent part. Finally, a numerical example for controlling a ship steering system is provided to demonstrate the validity and effectiveness of the proposed sliding mode fuzzy control method.

In this paper the comparison of a proposed neural network with generalized type-2 fuzzy weights (NNGT2FW) with respect to the monolithic neural network (NN) and the neural network with interval type-2 fuzzy weights (NNIT2FW) is presented. Generalized type-2 fuzzy inference systems are used to obtain the generalized type-2 fuzzy weights and are designed by a strategy of increasing and decreasing an epsilon variable for obtaining the different sizes of the footprint of uncertainty (FOU) for the generalized membership functions. The proposed method is based on recent approaches that handle weight adaptation using type-1 and type-2 fuzzy logic. The approach is applied to the prediction of the Mackey-Glass time series, and results are shown to outperform the results produced by other neural models. Gaussian noise was applied to the test data of the Mackey-Glass time series for finding out which of the presented methods in this paper shows better performance and tolerance to noise.

In this paper a new effective and scalable differential evolution algorithm is proposed for optimizing the Satellite Broadcast's Scheduling problem (SBS). The satellite broadcast's scheduling optimization problem is known to be an NP-complete problem in which the aim is to find a valid broadcasting pattern to earth-stationed terminals which maximizes the number of timeslots utilized for broadcasting under certain constraints. The algorithm proposed SD-BDE, is a binary version of Differential Evolution hybridized with ideas extracted from Stochastic Diffusion search. On top of that a repair heuristic mechanism is added to resolve constraint violations. Preliminary analysis shows that the repair heuristic is very effective as compared to other versions which include other heuristics. Further, the performance of the proposed algorithm is tested thoroughly against published work toward solving SBS problem as well as state-of-the-art existing binary-coded population-based similar algorithms. In this paper, we used, along with instances reported in the literature for such problem, randomly generated benchmark's instances of varying sizes for the sake of creating a unified large-scale set to compare different algorithm against. Experimental results show that the proposed algorithm outperformed the existing algorithms by finding better or optimal solutions for almost all tested benchmarks.

This paper develops a new method for solving multiple attribute group decision-making (MAGDM) problems with Atanassov's interval-valued intuitionistic fuzzy values (AIVIFVs) and incomplete attribute weight information. Firstly, we investigate the asymptotic property of the Atanassov's interval-valued intuitionistic fuzzy (AIVIF) matrix. It is demonstrated that after applying weights an infinite number of times, all elements in an AIVIF matrix will approach the same AIVIFV without regard to the initial values of elements. Then, the weight of each decision maker (DM) with respect to every attribute is determined by considering the similarity degree and proximity degree simultaneously. To avoid weighting an AIVIF matrix too many times, the collective decision matrix is transformed into an interval matrix using the risk coefficient of DMs. Subsequently, to derive the attribute weights objectively, we construct a multi-objective interval-programming model that is solved by transforming it into a linear program. The ranking order of alternatives is generated by the comprehensive interval values of alternatives. Finally, an example of a research and development (R&D) project selection problem is provided to illustrate the implementation process and applicability of the method developed in this paper.

Given a set of weighted objects in a data space, the MaxRS problem in spatial databases studied in a VLDB 2012 paper is to find a location for a rectangular region of a given size such that the weighted sum of all the objects covered by the rectangular region centered at the optimal location is maximized. This problem is useful in lots of location-based service applications, such as finding the location for a new fast food restaurant with a limited delivery range attracting the greatest number of customers. The existing MaxRS problem assumes that the rectangular region is always placed horizontally and is non-rotatable. However, under this assumption, the weighted sum of all the covered objects may not be the greatest when the rectangular region is rotatable. In this paper, we propose a generalized MaxRS problem called rotating MaxRS without this assumption. In rotating MaxRS, the rectangular region is rotatable and can be associated with an inclination angle. The goal of our problem is to find a location and an inclination angle such that the weighted sum of all the objects covered by the rectangular region of a given size centered at this location with this inclination angle is the greatest. We also present an efficient algorithm for the problem. Extensive experiments were conducted to verify the efficiency of our algorithms based on the real and synthetic datasets. The experimental results show that the weighted sum of all the objects in the rotating MaxRS queries can be increased with up to 300% on the synthetic datasets compared with existing non-rotating MaxRS queries, which shows the significance of the new rotating MaxRS queries.

Novel Evolutionary Algorithms are usually tested on sets of artificially-constructed benchmark problems. Such problems are often created to make the search of one global extremum (usually minimum) tricky. In this paper it is shown that benchmarking heuristics on either minimization or maximization of the same set of artificially-created functions (with equal bounds and number of allowed function calls) may lead to very different ranking of tested algorithms. As Evolutionary Algorithms and other heuristic optimizers are developed in order to be applicable to real-world problems, such result may raise doubts on the practical meaning of benchmarking them on artificial functions, as there is little reason that searching for the minimum of such functions should be more important than searching for their maximum.Thirty optimization heuristics, including a number of variants of Differential Evolution, as well as other kinds of Evolutionary Algorithms, Particle Swarm Optimization, Direct Search methods and - following the idea borrowed from No Free Lunch - pure random search are tested in the paper. Some discussion regarding the choice of the mean or the median performance for comparison is addressed and a short debate on the overall performance of particular methods is given.

In this paper, a sampled-data fuzzy control design approach for Takagi-Sugeno (T-S) model-based fuzzy systems is proposed. Firstly, the T-S model-based fuzzy systems are approximated by T-S fuzzy model-based discrete systems with parameter uncertainties. Then a sufficient condition on the existence of a sampled-data fuzzy controller for the fuzzy systems is formulated in the form of linear matrix inequalities which are proposed in some symmetrical form. So the proposed fuzzy control design approach will be less conservative. And some problems in the existing literature can be solved. Finally, a simulation example is discussed to show the effectiveness of the obtained fuzzy control design approach in this paper.

The purpose of this paper is to develop a linear programming technique for multidimensional analysis of preference (LINMAP) to address multiple criteria decision analysis problems within the interval type-2 fuzzy environment based on interval type-2 trapezoidal fuzzy numbers. Considering the issue of anchor dependency, we use multiple anchor points in the decision-making process and employ approximate positive-ideal and negative-ideal solutions as the points of reference. Selected useful properties of the approximate ideal solutions are also investigated. In contrast to the classical LINMAP methods, this paper directly generates approximate ideal solutions from the characteristics of all alternatives. Next, this work presents the concept of closeness-based indices using Minkowski distances with approximate ideal solutions to develop a new approach for determining measurements of consistency and inconsistency. Under incomplete preference information on paired comparisons of the alternatives, this paper provides a novel method that uses the concept of comprehensive closeness-based indices to measure the poorness of fit and the goodness of fit. By applying the consistency indices and inconsistency indices, this work formulates an optimization problem that can be solved for the optimal weights of the criteria and thus acquires the best compromise alternative. Additionally, this paper explores the problem of supplier selection and conducts a comparative discussion to validate the effectiveness and applicability of the proposed interval type-2 fuzzy LINMAP method with approximate ideal solutions. Furthermore, the proposed method is applied to address a marketplace decision difficulty (MPDD)-prone decision-making problem to provide additional contributions for practical implications.

It is well known that one of the basic advantages of evolutionary programming is in deciding the most suitable place for breeding offspring and finding the route toward the global minimum. To reach this goal, the algorithm needs to estimate the coordinates of the global minimum and then steer the new point toward it. In this paper, the estimation of the global minimum is calculated by weighted mean coordination of individuals (WMP), and then a road is mapped between the coordinates of the parents and the WMP. In the proposed method, fuzzy logic is used for deciding on the road and the best coordination to breed offspring. The proposed algorithm is tested on 65 well-known cost functions and is compared with five algorithms inside the EP family. In the next section of the paper, the algorithm is tested on the high-dimensional problem of modeling ozone layer data, which includes almost 26,000 unknown parameters. The results demonstrate the capability of the proposed method in having acceptable speed and accuracy.

Metaheuristics, such as evolutionary algorithms ( EA s), have been successfully applied to the problem of decision tree induction. Recently, an EA was proposed to evolve model trees, which are a particular type of decision tree that is employed to solve regression problems. However, there is a need to specialize the EA s in order to exploit the full potential of evolutionary induction. The main contribution of this paper is a set of solutions and techniques that incorporates knowledge about the inducing problem for the global model tree into the evolutionary search. The objective of this paper is to demonstrate that specialized EA can find more accurate and less complex solutions to the traditional greedy-induced counterparts and the straightforward application of EA .This paper proposes a novel solution for each step of the evolutionary process and presents a new specialized EA for model tree induction called the Global Model Tree ( GMT ). An empirical investigation shows that trees induced by the GMT are one order of magnitude less complex than trees induced by popular greedy algorithms, and they are equivalent in terms of predictive accuracy with output models from straightforward implementations of evolutionary induction and state-of-the-art methods.

The purpose of this paper is to develop a likelihood-based outranking method for handling multiple criteria decision analysis problems based on interval-valued intuitionistic fuzzy sets. Using the concept of likelihood of the interval-valued intuitionistic fuzzy preference relations, this paper determines certain generalized criteria and the corresponding likelihood-based preference functions. Based on some useful concepts of comprehensive preference indices, concordance indices, counter-likelihood-based preference functions, and discordance indices, this paper conducts a concordance-discordance analysis with concordance and discordance outranking relationships to determine a global Boolean matrix and acquire partial ranking orders of the alternatives. Alternatively, this paper determines complete ranking orders of the alternatives using the concepts of net concordance indices, net discordance indices, and mean outranking values. The feasibility and applicability of the proposed method are illustrated with a practical multiple criteria decision-making application concerning the selection of a suitable bridge construction method. Finally, comparative discussions with different decision-making methods are conducted to verify the effectiveness and advantages of the proposed method in aiding decision making.

Bayesian learning provides a firm theoretical basis of the design and exploitation of algorithms in data-streams processing (preprocessing, change detection, hypothesis testing, clustering, etc.). Primarily, it relies on a recursive parameter estimation of a firmly bounded complexity. As a rule, it has to approximate the exact posterior probability density (pd), which comprises unreduced information about the estimated parameter. In the recursive treatment of the data stream, the latest approximate pd is usually updated using the treated parametric model and the newest data and then approximated. The fact that approximation errors may accumulate over time course is mostly neglected in the estimator design and, at most, checked ex post. The paper inspects the estimator design with respect to the error accumulation and concludes that a sort of forgetting (pd flattening) is an indispensable part of a reliable approximate recursive estimation. The conclusion results from a Bayesian problem formulation complemented by the minimum Kullback-Leibler divergence principle. Claims of the paper are supported by a straightforward analysis, by elaboration of the proposed estimator to widely applicable parametric models and illustrated numerically.

Rough set theory provides a very useful idea of lower and upper approximations for inconsistent data. For incomplete data these approximations are not unique. In this paper we investigate properties of three well-known generalizations of approximations: singleton, subset and concept. These approximations were recently further generalized as to include an additional parameter α , interpreted as a probability. In this paper we report novel properties of singleton, subset and concept probabilistic approximations. Additionally, we validated such approximations experimentally. Our main objective was to test which of the singleton, subset and concept probabilistic approximations are the most useful for data mining. Our conclusion is that, for a given incomplete data set, all three approaches should be applied and the best approach should be selected as a result of ten-fold cross validation. Finally, we conducted experiments on complexity of rule sets and the total number of singleton, subset and concept approximations.

This paper proposes a neural network approach for efficiently solving general nonlinear convex programs with second-order cone constraints. The proposed neural network model was developed based on a smoothed natural residual merit function involving an unconstrained minimization reformulation of the complementarity problem. We study the existence and convergence of the trajectory of the neural network. Moreover, we show some stability properties for the considered neural network, such as the Lyapunov stability, asymptotic stability, and exponential stability. The examples in this paper provide a further demonstration of the effectiveness of the proposed neural network. This paper can be viewed as a follow-up version of [20,26] because more stability results are obtained.

The aim of this paper is to develop an ELECTRE (ELimination Et Choice Translating REality)-based outranking method for multiple criteria group decision-making within the environment of interval type-2 fuzzy sets. Along with considering the context of interval type-2 trapezoidal fuzzy numbers, this paper employs a hybrid averaging approach with signed distances to construct a collective decision matrix and proposes the use of ELECTRE-based outranking methods to analyze the collective interval type-2 fuzzy data. By applying a signed distance approach, this work identifies the concordance and discordance sets to determine the concordance and discordance indices, respectively, for each pair of alternatives. Based on an aggregate outranking matrix, a decision graph is constructed to determine the partial-preference ordering of the alternatives and the ELECTREcally non-outranked solutions. This paper provides additional approaches at the final selection stage to yield a linear ranking order of the alternatives. The feasibility and applicability of the proposed methods are illustrated with an example that addresses supplier selection, and a comparative analysis is performed with other approaches to validate the effectiveness of the proposed methodology.

In financial markets, some nonprobabilistic factors can be modeled as fuzzy numbers. Based on possibility theory and the assumption that the returns of assets are triangular fuzzy numbers, a bi-objective nonlinear portfolio selection model is proposed in this paper. This model aims to maximize the future expected return and minimize the future expected risk. Moreover, the obtained nonlinear bi-objective model is equivalent to the linear bi-objective minimizing programming model on the basis of possibilistic mean and possibilistic variance. Using the gradually tolerant constraint method proposed in this paper, we give a numerical example to illustrate the efficiency of the proposed model and method. The proposed method in this paper has improvements in two aspects. One is that our method offers several satisfactory solutions for the same model compared with the linear weighted method of Chang (2009), which offers only one satisfactory solution according to the investor's risk preference degree. The other is that the effective frontier of our method is more efficient than that of the method proposed by Markowitz (1987) and Zhang (2007).

Toksari, in his paper ‘Taylor series approach to fuzzy multiobjective linear fractional programming’, published in Information Sciences 178 (2008), proposed a new method to solve multiple objective linear fractional programming problems and declared that it is more effective when compare to previous methods. He constructed fuzzy goals and then used first order Taylor-polynomials to approximate the corresponding linear fractional membership functions by linear functions. Aggregating the linear functions Toksari obtained a crisp linear programming problem and claimed that it is ‘equal’ to the fuzzy fractional one. In this paper, we indicate the fallacy that arises by using Taylor approximation and propose an improved method that guarantees the efficiency of the solution it provides. Both practical applications from Toksari’s paper are recalled to show that the improvements we suggest are effective.

Differential Evolution (DE) is one of the most popular optimization methods for real-valued problems and a large number of its variants have been proposed so far. However, bringing together different ideas that already led to successful DE versions is rare in the literature. In the present paper a novel DE algorithm is proposed, in which three among the most efficient concepts already applied separately within DE framework are gathered together, namely: (1) the adaptation of algorithm control parameters and probabilities of using different mutation strategies; (2) the use of Nelder-Mead algorithm as a local search method hybridized with DE; and (3) the splitting mutation into Global and Local models, when Local mutation model is based on the concept of neighborhood of individuals organized on a ring topology. The performance of the novel algorithm, called Adaptive Memetic DE with Global and Local neighborhood-based mutation operators is compared with 13 different DE variants on a set of 25 popular problems which include rotated, shifted and hybrid composition functions. It is found that, although none DE algorithm outperforms all the others for the majority of problems, on average the proposed approach perform better than all 13 DE algorithms selected for comparison. The proposed algorithm is another heuristic approach developed to solve optimization problems. The question may arise, whether proposing novel methods is useful as No Free Lunch theorems for optimization state that the expected performance of all possible heuristics on all possible problems is equal. In the last section of the paper the limitations and implications of No Free Lunch theorems are discussed based on rich, but unfortunately frequently neglected literature. A very simple continuous and differentiable minimization problem is proposed, for which it is empirically verified that each among considered 14 DE algorithms perform poorer than random sampling. It is also empirically shown that when all considered DE algorithms search for the maximum of the proposed problem, they found lower minimum than DE algorithms searching for the minimum. Such result is not unexpected according to No Free Lunch theorems and should be considered as a precaution from generalization of good performance of heuristic optimization methods.

The TOPSIS method is a technique for establishing order preference by similarity to the ideal solution, and was primarily developed for dealing with real-valued data. This technique is currently one of most popular methods for Multiple Criteria Decision Making (MCDM). In many cases, it is hard to present precisely exact ratings of alternatives with respect to local criteria and as a result these ratings are seen as fuzzy values. A number of papers have been devoted to fuzzy extensions of the TOPSIS method in the literature, but these extensions are not complete since the ideal solutions are usually presented as real values (not by fuzzy values) or as fuzzy values which are not attainable in the decision matrix. In most of these papers, a defuzzification of elements of the fuzzy decision matrix is used, which leads inevitably to a loss of important information and may even produce the wrong results. In this paper, we propose a new direct approach to the fuzzy extension of the TOPSIS method which is free of the limitations of other known approaches. We show that the distances of the alternatives from the ideal solutions may be treated (in some sense) as modified weighted sums of local criteria. It is known that using weighted sums is not the best approach to the aggregation of local criteria in many real-world situations. Therefore, here, we propose the use, in addition to weighted sums, some other types of local criteria aggregation in the TOPSIS method and we develop a method for the generalization of different aggregation modes, providing compromised final results.

Interpolativity is one of the most important properties of a fuzzy inference system. It is well known that normal antecedent fuzzy sets forming a Ruspini partition constitute a practical setting ensuring interpolativity. In case of a fuzzy rule base expressing a monotone relationship, another desirable property is the monotonicity of the resulting function (after defuzzification). Unfortunately, this goal may often only be reached through the application of the at-least and/or at-most modifiers to the antecedent and consequent fuzzy sets. However, this approach does not seem compatible with the practical setting of a Ruspini partition. This paper shows that the situation is less conflicting than it seems, and that interpolativity can still be guaranteed, in the same practical setting, and, interestingly, from two different modeling points of view. This paper addresses the case of single-input single-output fuzzy rules.

Most real world multi-objective problems (MOPs) have a complicated solution space. Facing such problems, a direction vectors based co-evolutionary multi-objective optimization algorithm (DVCMOA) that introduces the decomposition idea from MOEA/D to co-evolutionary algorithms is proposed in this paper. It is novel in the sense that DVCMOA applies the concept of direction vectors to co-evolutionary algorithms. DVCMOA first divides the entire population into several subpopulations on the basis of the initial direction vectors in the objective space. Then, it solves MOPs through the co-evolutionary interaction among the subpopulations in which individuals are classified according to their direction vectors. Finally, it explores the less developed regions to maintain the relatively uniform distribution of the solution space. In this way, DVCMOA has advantages in convergence, diversity and uniform distribution of the non-dominated solution set, which are explained through comparison with other state-of-the-art multi-objective optimization evolutionary algorithms (MOEAs) in this paper. DVCMOA is shown to be effective on 6 multi-objective 0-1 knapsack problems.

This paper proposes a model for a robust shortest path problem with random edge costs. The proposed model is formulated as a bicriteria optimization problem to minimize the total cost of the path from a start node to an end node as well as to maximize the range of the confidence interval satisfying the condition that the total cost of any path in the region is less than a target value in terms of robustness. It is not always the case that a solution that simultaneously optimizes all of the objectives exists when the objectives conflict with each other. In this paper, fuzzy goals for both objectives and the aggregation function are introduced in a solution approach that considers the decision maker's preference and the subjectivity of her or his judgment in multicriteria decision making. The fuzzy bicriteria problem is transformed into a constrained shortest path problem. Furthermore, an exact algorithm based on Dijkstra's algorithm and a heuristic algorithm based on the Lagrange relaxation-based algorithm are developed for the constrained shortest path problem.

The research on optimal design of infinite-impulse response (IIR) filters based on optimization techniques has gained much attention in recent years. However, due to the limited performance of the applied optimization techniques, the orders of the filters, which can be obtained, are very low in the previous research. Memetic algorithms (MAs) are widely recognized to have better convergence capability than their conventional counterparts. However, the universality of the MAs, e.g. the ability of solving diverse kinds of digital IIR filter designs, is still limited. In this paper, we design a Two-Stage ensemble Memetic Algorithm (TSMA) framework to more appropriately synthesize the strengths of the evolutionary global search and local search techniques. In the first optimization stage, a competition is held among the candidate local search techniques. Its major idea is to choose the best local search technique and to obtain good initial state. Inheriting the good information of the first stage, the second optimization stage is to implement effective adaptive MA to pursue high-quality solution. The experimental studies presented in this paper contain three aspects: (1) the benefits of the TSMA framework are experimentally investigated by comparing TSMA with its sub-optimizers and recent effective evolutionary algorithms (EAs) on 26 test functions; then (2) TSMA is compared with 4 MAs on the CEC05 functions to comprehensively show the advantages of TSMA; and (3) the TSMA and 6 state-of-the-art algorithms are applied to design high-order digital infinite-impulse response (IIR) filters. The experimental results definitely demonstrate the excellent effectiveness, efficiency and reliability of TSMA on both function optimization and digital IIR filter design tasks.

Teaching-Learning-Based Optimization (TLBO) seems to be a rising star from amongst a number of metaheuristics with relatively competitive performances. It is reported that it outperforms some of the well-known metaheuristics regarding constrained benchmark functions, constrained mechanical design, and continuous non-linear numerical optimization problems. Such a breakthrough has steered us towards investigating the secrets of TLBO's dominance. This paper reports our findings on TLBO qualitatively and quantitatively through code-reviews and experiments, respectively. Our findings have revealed three important mistakes regarding TLBO: (1) at least one unreported but important step; (2) incorrect formulae on a number of fitness function evaluations; and (3) misconceptions about parameter-less control. Additionally, unfair experimental settings/conditions were used to conduct experimental comparisons (e.g., different stopping criteria). The experimental results for constrained and unconstrained benchmark functions under fairly equal conditions failed to validate its performance supremacy. The ultimate goal of this paper is to provide reminders for metaheuristics' researchers and practitioners in order to avoid similar mistakes regarding both the qualitative and quantitative aspects, and to allow fair comparisons of the TLBO algorithm to be made with other metaheuristic algorithms.

This paper presents an algorithm, named the uniform design multiobjective differential algorithm based on decomposition (UMODE/D), for optimizing multiobjective problems. The algorithm is a modification to the new version of MOEA/D based on differential evolution (DE), i.e., MOEA/D-DE proposed by Li and Zhang (2009) [20]. Its distinguishing features include: (1) The uniform design method is applied to generate the aggregation coefficient vectors so that the decomposed scalar optimization subproblems are uniformly distributed, and therefore the algorithm could explore uniformly the region of interest from the initial iteration; (2) The simplified quadratic approximation with three best points is employed to improve the local search ability and the accuracy of the minimum scalar aggregation function value. UMODE/D is compared with the original MOEA/D-DE and NSGA-II by solving a wide set of problems with complicated Pareto set shapes in this paper. Experimental results indicate that UMODE/D significantly outperforms MOEA/D-DE and NSGA-II on these test problems. Two sets of experiments are carried out to illustrate the efficiency of the uniform design method and the simplified quadratic approximation separately. In addition, UMODE/D is tested on CEC 2009 problems and combinatorial knapsack problems. Experimental results show that the proposed algorithm performs better than the further improved MOEA/D for almost all the CEC 2009 problems, and the results obtained are very competitive when comparing UMODE/D with some other algorithms on these multiobjective knapsack problems.

ross-entropy is a measure of the difference between two distribution functions. In order to deal with the divergence of uncertain variables via uncertainty distributions, this paper aims at introducing the concept of cross-entropy for uncertain variables based on uncertain theory, as well as investigating some mathematical properties of this concept. Several practical examples are also provided to calculate uncertain cross-entropy. Furthermore, the minimum cross-entropy principle is proposed in this paper. Finally, a study of generalized cross-entropy for uncertain variables is carried out.

The paper deals with the fuzzy extension of the weighted average operation. First, we study the convenient ways how uncertain weights and weighted values can be modeled by fuzzy vectors. We show that, in comparison to a tuple of fuzzy numbers that have been used for modeling uncertain values of particular weights and weighted values up to now, fuzzy vectors extend the possibilities of utilizing the vague expert information concerning the weights and weighted values. Next, we focus on computation of a fuzzy weighted average of a fuzzy vector of weighted values with a fuzzy vector of weights. We derive a general formula and we study its special forms. The advantage of the approach presented in the paper is that the resulting fuzzy weighted average is not overly imprecise since every available information about its variables is involved in computation. This fact is illustrated by several examples. Finally, we briefly discuss the problem of defuzzification of the resulting fuzzy weighted average.

This paper provides a continuation of the idea presented by Yin et al. [Yin et al., Some scheduling problems with general position-dependent and time-dependent learning effects, Inform. Sci. 179 (2009) 2416-2425]. For each of the following three objectives, total weighted completion time, maximum lateness and discounted total weighted completion time, this paper presents an approximation algorithm which is based on the optimal algorithm for the corresponding single-machine scheduling problem and analyzes its worst-case bound. It shows that the single-machine scheduling problems under the proposed model can be solved in polynomial time if the objective is to minimize the total lateness or minimize the sum of earliness penalties. It also shows that the problems of minimizing the total tardiness, discounted total weighted completion time and total weighted earliness penalty are polynomially solvable under some agreeable conditions on the problem parameters.

Research on the robustness of fuzzy logic systems (FLSs), an imperative factor in the design process, is very limited in the literature. Specifically, when a system is subjected to small deviations of the sampling points (operating points), it is of great interest to find the maximum tolerance of the system, which we refer to as the system's robustness. In this paper, we present a methodology for the robustness analysis of interval type-2 FLSs (IT2 FLSs) that also holds for T1 FLSs, hence, making it more general. A procedure for the design of robust IT2 FLSs with a guaranteed performance better than or equal to their T1 counterparts is then proposed. Several examples are performed to demonstrate the effectiveness of the proposed methodologies. It was concluded that both T1 and IT2 FLSs can be designed to achieve robust behavior in various applications, and preference one or the other, in general, is application-dependant. IT2 FLSs, having a more flexible structure than T1 FLSs, exhibited relatively small approximation errors in the several examples investigated. The methodologies presented in this paper lay the foundation for the design of FLSs with robust properties that will be very useful in many practical modeling and control applications.

In this paper an age-structured population growth model, based on a fuzzy cellular structure, is proposed. An age-structured population growth model enables a better description of population dynamics. In this paper, the dynamics of a particular bird species is considered. The dynamics is governed by the variation of natality, mortality and emigration rates, which in this work are evaluated using an interval type-2 fuzzy logic system. The use of type-2 fuzzy logic enables handling the effects caused by environment heterogeneity on the population. A set of fuzzy rules, about population growth, are derived from the interpretation of the ecological laws and the bird life cycle. The proposed model is formulated using discrete mathematics within the framework of a fuzzy cellular structure. The fuzzy cellular structure allows us to visualize the evolution of the population's spatial dynamics. The spatial distribution of the population has a deep effect on its dynamics. Moreover, the model enables not only to estimate the percentage of occupation on the cellular space when the species reaches its stable equilibrium level, but also to observe the occupation patterns.

This is the post-print version of the final paper published in Information Sciences. The published article is available from the link below. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. Copyright @ 2010 Elsevier B.V. The aggregation of objectives in multiple criteria programming is one of the simplest and widely used approach. But it is well known that this technique sometimes fail in different aspects for determining the Pareto frontier. This paper proposes a new approach for multicriteria optimization, which aggregates the objective functions and uses a line search method in order to locate an approximate efficient point. Once the first Pareto solution is obtained, a simplified version of the former one is used in the context of Pareto dominance to obtain a set of efficient points, which will assure a thorough distribution of solutions on the Pareto frontier. In the current form, the proposed technique is well suitable for problems having multiple objectives (it is not limited to bi-objective problems) and require the functions to be continuous twice differentiable. In order to assess the effectiveness of this approach, some experiments were performed and compared with two recent well known population-based metaheuristics namely ParEGO and NSGA II. When compared to ParEGO and NSGA II, the proposed approach not only assures a better convergence to the Pareto frontier but also illustrates a good distribution of solutions. From a computational point of view, both stages of the line search converge within a short time (average about 150 ms for the first stage and about 20 ms for the second stage). Apart from this, the proposed technique is very simple, easy to implement and use to solve multiobjective problems. CNCSIS IDEI 2412, Romania

This paper studies a renewal reward process with fuzzy random interarrival times and rewards under the @?-independence associated with any continuous Archimedean t-norm @?. The interarrival times and rewards of the renewal reward process are assumed to be positive fuzzy random variables whose fuzzy realizations are @?-independent fuzzy variables. Under these conditions, some limit theorems in mean chance measure are derived for fuzzy random renewal rewards. In the sequel, a fuzzy random renewal reward theorem is proved for the long-run expected reward per unit time of the renewal reward process. The renewal reward theorem obtained in this paper can degenerate to that of stochastic renewal theory. Finally, some application examples are provided to illustrate the utility of the result.

Most contemporary multi-objective evolutionary algorithms (MOEAs) store and handle a population with a linear list, and this may impose high computational complexities on the comparisons of solutions and the fitness assignment processes. This paper presents a data structure for storing the whole population and their dominating information in MOEAs. This structure, called a Dominance Tree (DT), is a binary tree that can effectively and efficiently store three-valued relations (namely dominating, dominated or non-dominated) among vector values. This paper further demonstrates DT's potential applications in evolutionary multi-objective optimization with two cases. The first case utilizes the DT to improve NSGA-II as a fitness assignment strategy. The second case demonstrates a DT-based MOEA (called a DTEA), which is designed by leveraging the favorable properties of the DT. The simulation results show that the DT-improved NSGA-II is significantly faster than NSGA-II. Meanwhile, DTEA is much faster than SPEA2, NSGA-II and an improved version of NSGA-II. On the other hand, in regard to converging to the Pareto optimal front and maintaining the diversity of solutions, DT-improved NSGA-II and DTEA are found to be competitive with NSGA-II and SPEA2.