Showing papers in "Fuzzy Optimization and Decision Making in 2014"

Uncertainty distribution and independence of uncertain processes

Abstract: Uncertain process is a sequence of uncertain variables indexed by time. This paper presents a concept of uncertainty distribution for describing uncertain process. Some sufficient and necessary conditions are also proved for uncertainty distribution and inverse uncertainty distribution of uncertain process. Finally, this paper proposes an independence definition of uncertain processes and shows some mathematical properties of it.

Fuzzy interaction regression for short term load forecasting

Abstract: Electric load forecasting is a fundamental business process and well-established analytical problem in the utility industry. Due to various characteristics of electricity demand series and the business needs, electric load forecasting is a classical textbook example and popular application field in the forecasting community. During the past 30 plus years, many statistical and artificial intelligence techniques have been applied to short term load forecasting (STLF) with varying degrees of success. Although fuzzy regression has been tried for STLF for about a decade, most research work is still focused at the theoretical level, leaving little value for practical applications. A primary reason is that inadequate attention has been paid to the improvement of the underlying linear model. This application-oriented paper proposes a fuzzy interaction regression approach to STLF. Through comparisons to three models (two fuzzy regression models and one multiple linear regression model) without interaction effects, the proposed approach shows superior performance over its counterparts. This paper also offers critical comments to a notable but questionable paper in this field. Finally, tips for practicing forecasting using fuzzy regression are discussed.

Multi-objective optimization in uncertain random environments

Abstract: Uncertain random variables are used to describe the phenomenon of simultaneous appearance of both uncertainty and randomness in a complex system. For modeling multi-objective decision-making problems with uncertain random parameters, a class of uncertain random optimization is suggested for decision systems in this paper, called the uncertain random multi-objective programming. For solving the uncertain random programming, some notions of the Pareto solutions and the compromise solutions as well as two compromise models are defined. Subsequently, some properties of these models are investigated, and then two equivalent deterministic mathematical programming models under some particular conditions are presented. Some numerical examples are also given for illustration.

Almost sure stability for uncertain differential equation

Abstract: Uncertain differential equation is a type of differential equation driven by Liu process. So far, concepts of stability and stability in mean for uncertain differential equations have been proposed. This paper aims at providing a concept of almost sure stability for uncertain differential equation. A sufficient condition is given for an uncertain differential equation being almost surely stable, and some examples are given to illustrate the effectiveness of the sufficient condition.

Universal integrals based on copulas

Abstract: A hierarchical family of integrals based on a fixed copula is introduced and discussed. The extremal members of this family correspond to the inner and outer extension of integrals of basic functions, the copula under consideration being the corresponding multiplication. The limits of the members of the family are just copula-based universal integrals as recently introduced in Klement et al. (IEEE Trans Fuzzy Syst 18:178---187, 2010). For the product copula, the family of integrals considered here contains the Choquet and the Shilkret integral, and it belongs to the class of decomposition integrals proposed in Even and Lehrer (Econ Theory, 2013) as well as to the class of superdecomposition integrals introduced in Mesiar et al. (Superdecomposition integral, 2013). For the upper Frechet-Hoeffding bound, the corresponding hierarchical family contains only two elements: all but the greatest element coincide with the Sugeno integral.

The α -cost minimization model for capacitated facility location-allocation problem with uncertain demands

Abstract: Facility location-allocation problem aims at determining the locations of some facilities to serve a set of spatially distributed customers and the allocation of each customer to the facilities such that the total transportation cost is minimized. In real life, the facility location-allocation problem often comes with uncertainty for lack of the information about the customers' demands. Within the framework of uncertainty theory, this paper proposes an uncertain facility location-allocation model by means of chance-constraints, in which the customers' demands are assumed to be uncertain variables. An equivalent crisp model is obtained via the $$\alpha $$ ? -optimistic criterion of the total transportation cost. Besides, a hybrid intelligent algorithm is designed to solve the uncertain facility location-allocation problem, and its viability and effectiveness are illustrated by a numerical example.

Uncertain agency models with multi-dimensional incomplete information based on confidence level

Abstract: This paper discusses a principal---agent problem with multi-dimensional incomplete information between a principal and an agent. Firstly, how to describe the incomplete information in such agency problem is a challenging issue. This paper characterizes the incomplete information by uncertain variable, because it has been an appropriate tool to depict subjective assessment and model human uncertainty. Secondly, the relevant literature often used expected-utility-maximization to measure the two participators' goals. However, Ellsberg paradox indicates that expected utility criterion is not always appropriate to be regarded as decision rule. For this reason, this paper presents another decision rule based on confidence level. Instead of expected-utility-maximization, the principal's aim is to maximize his potential income under the acceptable confidence level, and the agent's aim depends on whether he has private information about his efforts. According to the agent's different decision rules, three classes of uncertain agency (UA) models and their respective optimal contracts are presented. Finally, a portfolio selection problem is studied to demonstrate the modeling idea and the viability of the proposed UA models.

Stackelberg game models between two competitive retailers in fuzzy decision environment

Abstract: In this paper, we study the pricing problem in a fuzzy supply chain that consists of a manufacturer and two competitive retailers. There is a single product produced by a manufacturer and then sold by two competitive retailers to the consumers. The manufacturer acting as a leader determines the wholesale price, and the retailers acting as the followers set their sale prices independently. Both the manufacturing cost and the demand for product are characterized as fuzzy variables, we analyze how the manufacturer and the retailers make their pricing decisions with the duopolistic retailers' different behaviors: competition strategy and collusion strategy, and develop the expected value models in this paper. Finally, numerical examples illustrate the effectiveness of the proposed two-echelon models using fuzzy set theory.

Combining compound linguistic ordinal scale and cognitive pairwise comparison in the rectified fuzzy TOPSIS method for group decision making

Abstract: Group decision making is the process to explore the best choice among the screened alternatives under predefined criteria with corresponding weights from assessment of a group of decision makers The Fuzzy TOPSIS taking an evaluated fuzzy decision matrix as input is a popular tool to analyze the ideal alternative This research, however, finds that the classical fuzzy TOPSIS produces a misleading result due to some inappropriate definitions, and proposes the rectified fuzzy TOPSIS addressing two technical problems As the decision accuracy also depends on the evaluation quality of the fuzzy decision matrix comprising rating scores and weights, this research applies compound linguistic ordinal scale as the fuzzy rating scale for expert judgments, and cognitive pairwise comparison for determining the fuzzy weights The numerical case of a robot selection problem demonstrates the hybrid approach leading to the much reliable result for decision making, comparing with the conventional fuzzy Analytic Hierarchy Process and TOPSIS

On characterizing features of OWA aggregation operators

Abstract: We introduce the ordered weighted averaging (OWA) operator and emphasize how the choice of the weights, the weighting vector, allows us to implement different types of aggregation. We describe two important characterizing features associated with OWA weights. The first of these is the attitudinal character and the second is measure of dispersion. We discuss some methods for generating the weights and the role that these characterizing features can play in the determination of the OWA weights. We note that while in many cases these two features can help provide a clear distinction between different types of OWA operators there are some important cases in which these two characterizing features do not distinguish between OWA aggregations. In an attempt to address this we introduce a third characterizing feature associated with an OWA aggregation called the focus. We look at the calculation of this feature in a number of different situations.

The existence and uniqueness of fuzzy solutions for hyperbolic partial differential equations

Abstract: Fuzzy hyperbolic partial differential equation, one kind of uncertain differential equations, is a very important field of study not only in theory but also in application. This paper provides a theoretical foundation of numerical solution methods for fuzzy hyperbolic equations by considering sufficient conditions to ensure the existence and uniqueness of fuzzy solution. New weighted metrics are introduced to investigate the solvability for boundary valued problems of fuzzy hyperbolic equations and an extended result for more general classes of hyperbolic equations is initiated. Moreover, the continuity of the Zadeh's extension principle is used in some illustrative examples with some numerical simulations for $$\alpha $$ ? -cuts of fuzzy solutions.

Semideviations of reduced fuzzy variables: a possibility approach

Abstract: This paper proposes new methods to reduce the uncertain information embedded in the secondary possibility distribution of a type-2 fuzzy variable. Based on possibility measure, we define the lower value-at-risk (VaR) and upper VaR of a regular fuzzy variable, and develop the VaR-based reduction methods for type-2 fuzzy variables. The proposed VaR-based reduction methods generalize some existing reduction methods by introducing possibility level parameter in distribution functions. For VaR reduced fuzzy variables, we employ Lebesgue---Stieltjes (L---S) integral to define three $$n$$ n th semideviations to gauge the risk resulted from asymmetric fuzzy uncertainty. Furthermore, we compute the mean values and semideviations of the VaR reduced fuzzy variables, and derive some useful analytical expressions. The theoretical results obtained in this paper have potential applications in practical risk management and engineering optimization problems.

A semantic study of the first-order predicate logic with uncertainty involved

Abstract: In this paper, we provide a semantic study of the first-order predicate logic for situations involving uncertainty. We introduce the concepts of uncertain predicate proposition, uncertain predicate formula, uncertain interpretation and degree of truth in the framework of uncertainty theory. Compared with classical predicate formula taking true value in $$\{0,1\}$$ { 0 , 1 } , the degree of truth of uncertain predicate formula may take any value in the unit interval $$[0,1]$$ [ 0 , 1 ] . We also show that the uncertain first-order predicate logic is consistent with the classical first-order predicate logic on some laws of the degree of truth.

Computing fuzzy process efficiency in parallel systems

Abstract: This paper deals with parallel process systems in which the input and output data are fuzzy. The $$\upalpha $$ ? -level based approach is used to compute the fuzzy system efficiency and a simple procedure is proposed to estimate the fuzzy efficiency of the different processes. The main contribution of the paper is estimating the latter taking into account the variability of the process efficiencies compatible with a given value of the system efficiency. This variability comes from the existence of alternative optimal weights in the system efficiency multiplier network DEA models. The computation of the fuzzy system efficiency involves one Linear and one Non-linear Program for each $$\upalpha $$ ? -cut while the computation of each process efficiency requires solving just a couple of related Linear Programs for each $$\upalpha $$ ? -cut. The proposed approach is illustrated with a parallel systems dataset extracted from the literature.

Pessimistic, optimistic, and minimax regret approaches for linear programs under uncertainty

Abstract: Uncertain data appearing as parameters in linear programs can be categorized variously. This paper deals with merely probability, belief (necessity), plausibility (possibility), and random set information of uncertainties. However, most theoretical approaches and models limit themselves to the analysis involving merely one kind of uncertainty within a problem. Moreover, none of the approaches concerns itself with the fact that random set, belief (necessity), and plausibility (possibility) convey the same information. This paper presents comprehensive methods for handling linear programs with mixed uncertainties which also preserve all details about uncertain data. We handle mixed uncertainties as sets of probabilities which lead to optimistic, pessimistic, and minimax regret in optimization criteria.

Estimation of mutual information by the fuzzy histogram

Abstract: Mutual Information (MI) is an important dependency measure between random variables, due to its tight connection with information theory. It has numerous applications, both in theory and practice. However, when employed in practice, it is often necessary to estimate the MI from available data. There are several methods to approximate the MI, but arguably one of the simplest and most widespread techniques is the histogram-based approach. This paper suggests the use of fuzzy partitioning for the histogram-based MI estimation. It uses a general form of fuzzy membership functions, which includes the class of crisp membership functions as a special case. It is accordingly shown that the average absolute error of the fuzzy-histogram method is less than that of the naive histogram method. Moreover, the accuracy of our technique is comparable, and in some cases superior to the accuracy of the Kernel density estimation (KDE) method, which is one of the best MI estimation methods. Furthermore, the computational cost of our technique is significantly less than that of the KDE. The new estimation method is investigated from different aspects, such as average error, bias and variance. Moreover, we explore the usefulness of the fuzzy-histogram MI estimator in a real-world bioinformatics application. Our experiments show that, in contrast to the naive histogram MI estimator, the fuzzy-histogram MI estimator is able to reveal all dependencies between the gene-expression data.

Ranking probability measures by inclusion indices in the case of unknown utility function

Abstract: This paper gives a way of analyzing decisions in the case of unknown utility function, or more precisely, when we know only a linear order on an income space. It is shown that in this situation, decisions and corresponding probability measures are partially ordered, and this order is identical to the inclusion relation of comonotone fuzzy sets. It enables us to use inclusion indices of fuzzy sets to analyze the comparability of decisions. To do this, we introduce an inclusion index having properties, which are close to ones of the classical expected utility.

Duality for a class of fuzzy nonlinear optimization problem under generalized convexity

Abstract: In this paper, a Mond-Weir type dual program for a nonlinear primal problem under fuzzy environment is formulated. The solution concept of primal-dual problems is inspired by the nondominated solution. We have considered ordering among fuzzy numbers as a partial ordering and using the concept of Hukuhara difference between two fuzzy numbers and $$H$$ H -differentiability, appropriate duality theorems are established under pseudo/quasi-convexity assumptions. We have also illustrated a numerical example which satisfies the duality relations discussed in the paper.

Dynamic resource allocation in fuzzy coalitions: a game theoretic model

Abstract: We introduce an efficient and dynamic resource allocation mechanism within the framework of a cooperative game with fuzzy coalitions (cooperative fuzzy game). A fuzzy coalition in a resource allocation problem can be so defined that membership grades of the players in it are proportional to the fractions of their total resources. We call any distribution of the resources possessed by the players, among a prescribed number of coalitions, a fuzzy coalition structure and every membership grade (equivalently fraction of the total resources), a resource investment. It is shown that this resource investment is influenced by the satisfaction of the players in regard to better performance under a cooperative setup. Our model is based on the real life situations, where possibly one or more players compromise on their resource investments in order to help forming coalitions.

Integrated inventory problem under trade credit in fuzzy random environment

Abstract: This paper is concerned with an integrated inventory problem under trade credit where both the demand rate and deteriorating rate are assumed to be uncertain and characterized as fuzzy random variables with known distributions. The objective of this paper is to determine the optimal inventory policy by optimizing simultaneously the replenishment cycle length and trade credit period. At first, three decision criteria are given: (1) expected value criterion, (2) chance-constrained criterion and (3) chance maximization criterion. Then, after building the fuzzy random models based on the above decision criterion, a hybrid intelligent algorithm by integrating fuzzy random simulation and genetic algorithm is employed to deal with these models. At the end, three numerical examples are given to illustrate the benefits of the models and show the effectiveness of the algorithms.

On Arrow---Sen style equivalences between rationality conditions for fuzzy choice functions

Abstract: The Arrow---Sen Theorem establishes the equivalence between different rationality conditions for a choice function. In this contribution we deal with fuzzy versions of these rationality conditions and we study the connection between them. We recall results found in the literature and we prove that they are valid under weaker conditions. We also present new implications and counterexamples that show that our results cannot be obtained under weaker conditions.

Blackwell's theorem for fuzzy variables

Abstract: Recently, Zhao et al. (Euro J Oper Res 169:189---201, 2006) discussed a random fuzzy renewal process based on the random fuzzy theory and established Blackwell's theorem in random fuzzy sense. They obtained Blackwell's theorem for fuzzy variables by degenerating the process. However, this result is invalid. We provide some counterexamples and offer a corrected version of fuzzy Blackwell's theorem.

Renewal reward process for $$T$$T-related fuzzy random variables on $$(\mathbb {R}^{p}, \mathbb {R}^{q})$$(Rp,Rq)

Abstract: In this paper, following our previous studies, we investigate the renewal rewards process with respect to the necessity, credibility, chance measure and the expected value in which the random inter-arrival times and random rewards are characterized as weighted fuzzy numbers under $$t$$ t -norm-based fuzzy operations on $$\mathbb {R}^{p}$$ R p and $$\mathbb {R}^{q}\,\,p,\,q \ge 1,$$ R q p , q ? 1 , respectively. Many versions of $$T$$ T -related fuzzy renewal rewards theorems are proved by using the law of large numbers for weighted fuzzy variables on $$\mathbb {R}^{p}$$ R p . An application example is provided to illustrate the utility of the results.

Poincáre recurrence theorem in regular uncertain dynamic system

Abstract: Poincare recurrence theorem in an uncertain dynamic system is proved in the framework of uncertainty theory, which claims that almost every point of an uncertain event with positive uncertain measure will iterate back to the event for infinite times. This recurrence behaviour can be used to develop new results of uncertain variable in an uncertain dynamic system.