Entropy measures and granularity measures for interval and set-valued information systems

In many practical situations, some of the attribute values for an object may be interval and set-valued. The interval and set-valued information systems have been introduced. According to the semantic relation of attribute values, interval and set-valued information systems can be classified into two categories, disjunctive (type 1) and conjunctive (type 2) systems. This paper mainly focuses on semantic interpretation of type 1. Then, a new fuzzy preference relation for interval and set-valued information systems is defined. Moreover, based on the new fuzzy preference relation, the concepts of fuzzy information entropy, fuzzy rough entropy, fuzzy knowledge granulation and fuzzy granularity measure are studied and relationships between entropy measures and granularity measures are investigated. Finally, an illustrative example to substantiate the theoretical arguments is given. These results may supply a further understanding of the essence of uncertainty in interval and set-valued information systems.     

Similarity of binary relations based on <Emphasis Type="Italic">L</Emphasis>-fuzzy topologies

Binary relations play an important role in rough set theory. This paper investigates the similarity of binary relations based on L-fuzzy topologies, where L is a boolean algebra. First, rough approximations based on a boolean algebra are proposed through successor neighborhoods on binary relations. Next, L-fuzzy topologies induced by binary relations are investigated. Finally, similarity of binary relations is introduced by using the L-fuzzy topologies and the fact that every binary relation is solely similar to some preorder relation is proved. It is worth mentioning that similarity of binary relations are both originated in the L-fuzzy topology and independent of the L-fuzzy topology.     

A theoretical extension on the operational law for monotone functions of uncertain variables

The classical operational law of uncertain variables proposed by Liu makes an important contribution to the development of the uncertainty theory in both theories and applications. It provides a powerful and practical approach for calculating the uncertainty distribution of strictly monotone function of uncertain variables. However, the restriction on strictly monotone functions of the operational law limits its applications since many practical problems cannot be modeled by strictly monotone functions but general monotone functions. Therefore, an extension of the original operational law is needed. For this purpose, some properties concerning the uncertainty distributions of monotone functions of uncertain variables as well as the generalized inverse uncertainty distributions are presented first in this paper. On the basis of these discussions, a generalized operational law is proposed as a natural extension of the original operational law. Then the uncertainty distribution of a general monotone function of independent regular uncertain variables can be derived, which is analogous to the way that suggested by the original operational law for dealing with strictly monotone functions. Furthermore, as an application of the generalized operational law, a theorem for calculating the expected values of general monotone functions of uncertain variables is presented as well.     

Constrained differential evolution using generalized opposition-based learning

Differential evolution (DE) is a well-known optimization approach to deal with nonlinear and complex optimization problems. However, many real-world optimization problems are constrained problems that involve equality and inequality constraints. DE with constraint handling techniques, named constrained differential evolution (CDE), can be used to solve constrained optimization problems. In this paper, we propose a new CDE framework that uses generalized opposition-based learning (GOBL), named GOBL-CDE. In GOBL-CDE, firstly, the transformed population is generated using general opposition-based learning in the population initialization. Secondly, the transformed population and the initial population are merged and only half of the best individuals are selected to compose the new initial population to proceed mutation, crossover, and selection. Lastly, based on a jumping probability, the transformed population is calculated again after generating new populations, and the fittest individuals are selected to compose new population from the union of the current population and the transformed population. The GOBL-CDE framework can be applied to most CDE variants. As examples, in this study, the framework is applied to two popular representative CDE variants, i.e., rank-iMDDE and \(\varepsilon \)DEag. Experiment results on 24 benchmark functions from CEC’2006 and 18 benchmark functions from CEC’2010 show that the proposed framework is an effective approach to enhance the performance of CDE algorithms.     

Chain-reaction solution update in MOEA/D and its effects on multi- and many-objective optimization

MOEA/D is one of the promising evolutionary algorithms for multi- and many-objective optimization. To improve the search performance of MOEA/D, this work focuses on the solution update method in the conventional MOEA/D and proposes its alternative, the chain-reaction solution update. The proposed method is designed to maintain and improve the variable (genetic) diversity in the population by avoiding duplication of solutions in the population. In addition, the proposed method determines the order of existing solutions to be updated depending on the location of each offspring in the objective space. Furthermore, when an existing solution in the population is replaced by a new offspring, the proposed method tries to reutilize the existing solution for other search directions by recursively performing the proposed chain-reaction update procedure. This work uses discrete knapsack and continuous WFG4 problems with 2–8 objectives. Experimental results using knapsack problems show the proposed chain-reaction update contributes to improving the search performance of MOEA/D by enhancing the diversity of solutions in the objective space. In addition, experimental results using WFG4 problems show that the search performance of MOEA/D can be further improved using the proposed method.     

Genetic programming for QoS-aware web service composition and selection

Web services, which can be described as functionality modules invoked over a network as part of a larger application are often used in software development. Instead of occasionally incorporating some of these services in an application, they can be thought of as fundamental building blocks that are combined in a process known as Web service composition. Manually creating compositions from a large number of candidate services is very time consuming, and developing techniques for achieving this objective in an automated manner becomes an active research field. One promising group of techniques encompasses evolutionary computing, which can effectively tackle the large search spaces characteristic of the composition problem. Therefore, this paper proposes the use of genetic programming for Web service composition, investigating three variations to ensure the creation of functionally correct solutions that are also optimised according to their quality of service. A variety of comparisons are carried out between these variations and two particle swarm optimisation approaches, with results showing that there is likely a trade-off between execution time and the quality of solutions when employing genetic programming and particle swarm optimisation. Even though genetic programming has a higher execution time for most datasets, the results indicate that it scales better than particle swarm optimisation.     

A projection-based decomposition for the scalability of evolvable hardware

Scalability is a main and urgent problem in evolvable hardware (EHW) field. For the design of large circuits, an EHW method with a decomposition strategy is able to successfully find a solution, but requires a large complexity and evolution time. This study aims to optimize the decomposition on large-scale circuits so that it provides a solution for the EHW method to scalability and improves the efficiency. This paper proposes a projection-based decomposition (PD), together with Cartesian genetic programming (CGP) as an EHW system namely PD-CGP, to design relatively large circuits. PD gradually decomposes a Boolean function by adaptively projecting it onto the property of variables, which makes the complexity and number of sub-logic blocks minimized. CGP employs an evolutionary strategy to search for the simple and compact solutions of these sub-blocks. The benchmark circuits from the MCNC library, \(n\)-parity circuits, and arithmetic circuits are used in the experiment to prove the ability of PD-CGP in solving scalability and efficiency. The results illustrate that PD-CGP is superior to 3SD-ES in evolving large circuits in terms of complexity reduction. PD-CGP also outperforms GDD+GA in evolving relatively large arithmetic circuits. Additionally, PD-CGP successfully evolves larger \(n\)-even-parity and arithmetic circuits, which have not done by other approaches.     

Minimal attribute reduction with rough set based on compactness discernibility information tree

Minimal attribute reduction plays an important role in rough set. Heuristic algorithms are proposed in literature reviews to get a minimal reduction and yet an unresolved issue is that many redundancy non-empty elements involving duplicates and supersets exist in discernibility matrix. To be able to eliminate the related redundancy and pointless elements, in this paper, we propose a compactness discernibility information tree (CDI-tree). The CDI-tree has the ability to map non-empty elements into one path and allow numerous non-empty elements share the same prefix, which is recognized as a compact structure to store non-empty elements in discernibility matrix. A complete algorithm is presented to address Pawlak reduction based on CDI-tree. The experiment results reveal that the proposed algorithm is more efficient than the benchmark algorithms to find out a minimal attribute reduction.     

An adaptive support vector regressor controller for nonlinear systems

In this study, a novel online support vector regressor (SVR) controller based on system model estimated by a separate online SVR is proposed. The main idea is to obtain an SVR controller based on an estimated model of the system by optimizing the margin between reference input and system output. For this purpose, “closed-loop margin” which depends on tracking error is defined, then the parameters of the SVR controller are optimized so as to optimize the closed-loop margin and minimize the tracking error. In order to construct the closed-loop margin, the model of the system estimated by an online SVR is utilized. The parameters of the SVR controller are adjusted via the SVR model of system. The stability of the closed-loop system has also been analyzed. The performance of the proposed method has been evaluated by simulations carried out on a continuously stirred tank reactor (CSTR) and a bioreactor, and the results show that SVR model and SVR controller attain good modeling and control performances.     

Chaotic feature analysis and forecasting of Liujiang River runoff

Because most of runoff time series with limited amount of data reveal inherently nonlinear and stochastic characteristics and tend to show chaotic behavior, strategies based on chaotic analysis are popular methods to analyze them from real systems in nonlinear dynamics. Only one kind of predicted method for yearly rainfall-runoff forecasting cannot achieve perfect performance. Thus, a mixture strategy denoted by WT-PSR-GA-NN, which is composed of wavelet transform (WT), phase space reconstruction (PSR), neural network (NN) and genetic algorithm (GA), is presented in this paper. In the WT-PSR-GA-NN framework, the process to deal with time series gathered from Liujiang River runoff data is given as follows: (1) the runoff time series was first decomposed into low-frequency and high-frequency sub-series by wavelet transformation; (2) the two sub-series were separately and independently reconstructed into phase spaces; (3) the transformed time series in the reconstructed phase spaces were modeled by neural network, which is trained by genetic algorithm to avoid trapping into local minima; (4) the predicted results in low-frequency parts were combined with the ones of high-frequency parts, and reconstructed with wavelet inverse transformation, to form the future behavior of the runoff. Experiments show that WT-PSR-GA-NN is effective and its forecasting results are high in accuracy not only for the short-term yearly hydrological time series but also for the long-term one. The comparison results revealed that the overall forecasting performance of WT-PSR-GA-NN proposed by us is superior to other popularity methods for all the test cases. We can conclude that WT-PSR-GA-NN can not only increase the forecasted accuracy, but also its own competitiveness in efficiency, effectiveness and robustness.