Abstract: Uncertain fractional differential equations have been playing an important role in modelling complex dynamic systems. Early researchers have presented the extreme value theorems and time integral theorem on uncertain fractional differential equation. As applications of these theorems, this paper investigates the pricing problems of American option and Asian option under uncertain financial markets based on uncertain fractional differential equations. Then the analytical solutions and numerical solutions of these option prices are derived, respectively. Finally, some numerical experiments are performed to verify the effectiveness of our results. PubDate: 2021-03-05

Abstract: Group decision making (GDM) can be defined as an environment where there exist a set of possible alternatives and a set of individuals (experts, judgements, etc.). Preference relation is one of the most widely used preference representation structures in GDM. Considering that the self-confidence degree is an important part to express preference information, and double hierarchy linguistic preference relation (DHLPR) is a cognitive complex linguistic information representation tool to express complex linguistic information, this paper presents a novel preference relation named as self-confident DHLPR. In addition, a weight-determining method is developed, which considers three kinds of information including the subjective weights and two kinds of objective weights. Furthermore, a consensus model is set up to manage the GDM problems with self-confident DHLPRs based on the priority ordering theory. The effectiveness of the proposed consensus model is illustrated by a case study concerning the selection of optimal hospitals in the field of Telemedicine. Finally, a simulation experiment is devised to testify the proposed consensus model and then some comparisons with other consensus reaching models are provided from three different angles including the number of iterations, the consensus success ratio and the distance between the original and adjusted preferences. PubDate: 2021-03-01

Abstract: Uncertain differential equation is a type of differential equation driven by Liu process that is the counterpart of Wiener process in the framework of uncertainty theory. The stability theory is of particular interest among the properties of the solutions to uncertain differential equations. In this paper, we introduce the Lyapunov’s second method to study stability in measure and asymptotic stability of uncertain differential equation. Different from the existing results, we present two sufficient conditions in sense of Lyapunov stability, where the strong Lipschitz condition of the drift is no longer indispensable. Finally, illustrative examples are examined to certify the effectiveness of our theoretical findings. PubDate: 2021-03-01

Abstract: The challenge of complex multi-attribute large group decision-making (CMALGDM) is reflected from three perspectives: interrelated attributes, large group decision makers (DMs) and dynamic decision environments. However, there are few decision techniques that can address the three perspectives simultaneously. This paper proposes a random intuitionistic fuzzy factor analysis model, aiming to address the challenge of CMALGDM from the three perspectives. The proposed method effectively reduces the dimensionality of the original data and takes into account the underlying random environmental factors which may affect the performances of alternatives. The development of this method follows three steps. First, the random intuitionistic fuzzy variables are developed to deal with a hybrid uncertain situation where fuzziness and randomness co-exist. Second, a novel factor analysis model for random intuitionistic fuzzy variables is proposed. This model uses specific mappings or functions to define the way in which evaluations are affected by the dynamic environment vector through data learning or prior distributions. Third, multiple correlated attribute variables and DM variables are transformed into fewer independent factors by a two-step procedure using the proposed model. In addition, the objective classifications and weights for attributes and DMs are obtained from the results of orthogonal rotated factor loading. An illustrative case and detailed comparisons of decision results in different environmental conditions are demonstrated to test the feasibility and validity of the proposed method. PubDate: 2021-03-01

Abstract: In the data security context, anomaly detection is a branch of intrusion detection that can detect emerging intrusions and security attacks. A number of anomaly detection systems (ADSs) have been proposed in the literature that using various algorithms and techniques try to detect the intrusions and anomalies. This paper focuses on the ADS schemes which have applied fuzzy logic in combination with other machine learning and data mining techniques to deal with the inherent uncertainty in the intrusion detection process. For this purpose, it first presents the key knowledge about intrusion detection systems and then classifies the fuzzy ADS approaches regarding their utilized fuzzy algorithm. Afterward, it summarizes their major contributions and illuminates their advantages and limitations. Finally, concluding issues and directions for future researches in the fuzzy ADS context are highlighted. PubDate: 2021-03-01

Abstract: A behavioral multi-attribute decision making (BMADM) problem with probabilistic-based expressions is studied by considering decision-maker’s (DM) risk attitude and pre-evaluation. With consideration of information expressions for uncertainty, probabilistic interval numbers (PINs) and probabilistic linguistic terms (PLTs) are utilized to depict pre-evaluation information with respect to quantitative and qualitative attributes, respectively. Then surrounding the two kinds of probabilistic-based expressions, we propose a BMADM method with DM’s risk attitude being included based on regret theory. First, through taking into account characteristics of risk, we develop a basic utility function and a regret–rejoice function by considering risk-averse, risk-neutral and risk-seeking preference coefficients. Second, risk-based utility functions are examined for measuring PINs and PLTs. The third element is the establishment of optimization models for handling probability incompleteness to fully utilize the information. In the fourth step, a weighted comprehensive risk-based utility measurement is presented as a basis for making a selection. The final phase of the research is the application of the proposed method to one case, along with sensitivity and comparative analyses, as a means of illustrating the applicability and feasibility of the new method. PubDate: 2021-03-01

Abstract: As applications of the uncertainty theory to finance, uncertain stock models have been presented to describe the prices of stocks strongly influenced by human uncertainty. So far, large progress has been achieved on pricing problems of path-independent options of the uncertain stock models. This paper investigates a type of path-dependent exotic options of an uncertain stock model which are named barrier options. Pricing formulas are derived based on the structure of the solutions of uncertain differential equations, and numerical algorithms are designed to calculate the prices of the barrier options based on these formulas. PubDate: 2021-03-01

Abstract: Nonparametric regression analysis is a useful method to explore the relationships among the variables when a parametric form is not known. Assuming the observations of the model are imprecise and modeling the observed data via uncertain variables, this paper proposes least squares estimation of uncertain nonparametric regression model to explore the functional relationships between response variable and explanatory variable. In particular, we employ B-Splines and local polynomials to approximate the nonparametric function, respectively. Estimation of unknown function can be obtained as a solution of least squares and quadratic programming algorithm can be used to compute efficiently the estimator. Numerical examples are given to illustrate the proposed methods. PubDate: 2021-02-17

Abstract: This paper develops a unified and structured solution framework for the minimum spanning tree (MST) problem and its variants (e.g., constrained MST problem and inverse MST problem) on networks with fuzzy link weights. It is applicable to any additive decision criterion under fuzziness (e.g., expected value, value at risk, and conditional value at risk), for generalized cases that the link weights may be represented by arbitrary types of fuzzy variables. It also applies to the entropy criterion while the link weights are continuous fuzzy variables. Following the optimality conditions of the fuzzy MST under different decision criteria proved first in this paper, it is shown that the MST problem and its variants on a fuzzy network can be converted into equivalent deterministic counterparts on their corresponding crisp networks. Consequently, these problems can be effectively solved via their deterministic counterparts without fuzzy simulation, and meanwhile, the performance of the trees under a specified criterion is precisely measured. The accuracy and efficiency are both significantly improved compared with other fuzzy simulation-based approaches. Numerical examples illustrate the superiority of the proposed solution framework. Furthermore, some new theoretical conclusions on the MST problem under fuzziness are also presented. PubDate: 2021-02-15

Abstract: Uncertain differential game models conflicts and interests among players in the context of an uncertain dynamic system. However, cooperative behavior in uncertain differential game is an unexplored terrain. This paper proposes a spectrum of a cooperative uncertain differential game with transferrable payoffs. First, group rationality is achieved by maximizing the coalitional payoff through an uncertain optimal control method. Second, the concept of imputation is introduced as a solution, and a stability condition called subgame consistency condition for an imputation is proposed as well. Furthermore, the derivation of payoff distribution procedure for subgame consistent imputation is discussed. In addition, two optimal principles, i.e., Nash bargaining solution and Shapley value, are extended to this kind of model and are proved to be subgame consistent imputations, and their payoff distribution procedures are derived analytically. Finally, a two-person resource extraction game is studied for illustrating purpose. PubDate: 2021-02-13

Abstract: Evaluating and selecting suitable sustainable recycling partners is a key work in sustainable supply chain management. In order to deal with the probabilistic linguistic influence relations between criteria and obtain the key factors that influence the evaluation results of sustainable recycling partners, we propose a new decision-making trial and evaluation laboratory (DEMATEL) method. First, we propose a new generalized weighted ordered weighted averaging (GWOWA) operator and discuss its properties. Second, we use probabilistic linguistic term sets (PLTSs) to aggregate the experts’ hesitant fuzzy linguistic decision-making information and develop a novel method of transforming PLTSs into triangular fuzzy numbers (TFNs) based on the proposed GWOWA operator and the characteristics of PLTSs. Furthermore, we propose a method of making criteria relation analysis based on DEMATEL with TFNs. With the method, we not only access the importance weights of criteria but also obtain the influence relation among the criteria and cluster the criteria into two groups: cause group and effect group. Finally, we apply our method to a real case of sustainable recycling partner selection. PubDate: 2020-12-01 DOI: 10.1007/s10700-020-09326-9

Abstract: The aim of the paper is to solve the group decision making problems which contain inconsistent probabilistic linguistic preference relations (PLPRs) and unknown expert weights. When the PLPRs are inconsistent, there are contradictories in the preference relations expressed by the experts. The evaluation value with contradictory information will bring out an incorrect consequence in decision making. Hence, this paper develops a novel consistency measure method to gauge the consistency level of PLPRs. Moreover, a nonlinear optimization model is newly constructed to optimize the inconsistent PLPRs. The proposed methods overcome the limitations in the existing methods and ameliorate the interpretation and complexities of inconsistency PLPRs revise strategies. Additionally, a weighting method using fuzzy cooperative games with PLPRs is put forward to derive the weight vector of experts. It helps to balance the deviations between the individual PLPRs and the group PLRP. At last, a numerical example illustration for physician selection is put forward to demonstrate the effectiveness of the proposed model and its practical applicability. The comparative analysis gives deep insights into the rationality of the proposed model. PubDate: 2020-12-01 DOI: 10.1007/s10700-020-09329-6

Abstract: As a representative of the new economy, the web celebrity economy has achieved significant development in China with the rapid development of information technology and the Internet. In this environment, web celebrity shops encounter fierce business competition of peer competitors. Online reviews which imply the consumers’ attitudes and sentiments give the web celebrity shops good feedback to improve their competitiveness. Thus, taking milk tea as an example, this paper deeply investigates the assessment of web celebrity shops by mining online review. At the same time, we also discuss the competitive analysis and propose the corresponding improvement advices. In order to obtain the satisfaction assessments of web celebrity shops, on the one hand, we analyze topic extraction with latent dirichlet allocation (LDA) and determine the attributes that customers care about. On the other hand, we utilize long short-term memory (LSTM) and probabilistic linguistic term sets (PLTSs) to more precisely portray customers’ sentiment towards different attributes. By using fuzzy cognitive map (FCM) and the association rule, we further investigate the interrelationship among the attributes and construct the relationship graph between attributes for web celebrity shops. With the above results, we aggregate the decision information by designing improved extended Bonferroni mean (EBM) and obtain comprehensive evaluations. General speaking, this paper successfully transforms the unstructured data of online reviews into quantitative information and obtain satisfaction evaluations. With the aid of PLTSs and FCM, we further investigate the competitive analysis and propose improvement advices for each shop, which systematically provides us with a data-driven decision-making analysis model. PubDate: 2020-12-01 DOI: 10.1007/s10700-020-09327-8

Abstract: In this paper, we study four projection-based normalization models and a decision-making method for probabilistic linguistic multi-criteria decision-making problems, in which the assessment information about an alternative with respect to a criterion is incomplete and the criteria weight values are not precisely known but the ranges are available. To apply the projection to the probabilistic linguistic environment, we propose the equivalent expression forms of the probabilistic linguistic term sets, and then the equivalent transformation functions between the probabilistic linguistic term set and its associated vector are presented to realize the conversion between the operations on the probabilistic linguistic term sets and the operations on their associated vectors. Next, the projection formulas of the probabilistic linguistic term sets are introduced to build different normalization models for different types of uncertain probabilistic linguistic multi-criteria decision-making problems. After that, a new deviation degree formula is proposed to account for the rationality and validity of the normalization models from the theoretical perspective. Finally, the probabilistic linguistic two-step method is used to determine the criteria weights values and rank the alternatives, and the validity of these projection-based normalization models and our proposed decision-making method are illustrated by a case about the performance assessment of data hiding techniques. PubDate: 2020-12-01 DOI: 10.1007/s10700-020-09325-w

Abstract: When expressing preferences with different probability weights for different linguistic terms, only partial assessment information is usually to be provided. Then the probability information can be normalized to the interval probability, hence, using interval probabilistic linguistic term sets (IPLTs) is more appropriate. Considering this situation, interval probabilistic linguistic preference relation (IPLPR) is proposed. To measure the consistency of IPLPR, the consistency definition of IPLPR is put forward. For the consistent IPLPR, from which an expected consistent PLPR can be obtained, we can obtain interval weights as the final priorities by using the pairs of linear programming models. We also create the probabilistic linguistic geometric consistency index (PLGCI) of PLPRs to judge whether the IPLPR is satisfactorily consistent. For an unsatisfied consistency IPLPR, the adjusting algorithm is proposed. Probability information is firstly considered to be adjusted. If it is not possible to achieve satisfactory consistency through the adjustment of probability information, then the linguistic terms will be adjusted. In addition to examples of different situations, such as the consistency, satisfactory consistency and consistency improvement, the application example is also given to show the practicability of the proposed methods. PubDate: 2020-12-01 DOI: 10.1007/s10700-020-09328-7

Abstract: The probabilistic linguistic term set (PLTS) is a powerful tool for describing linguistic evaluations derived from expert teams and has adequate capability to identify preferences among different evaluations. Due to the practicability of PLTSs, probabilistic linguistic decision making problems have been widely investigated in recent years. However, no study on probabilistic linguistic outranking relations has been conducted. This study aims to explore effective processing for the complex two-dimension structure of PLTSs and formulate probabilistic linguistic dominance and opposition relations for multi-criteria sorting decision making. Linguistic scale functions, which can generate different semantics for linguistic variables under different decision making environments, are introduced to deal with the linguistic terms in PLTSs. In this way, the probabilistic linguistic dominance degree, concordance and discordance indices are defined by systematically comparing the probabilities of PLTSs. Then, two kinds of outranking relations with dominance and opposition for PLTSs are formulated based on the defined outranking indices. Subsequently, an innovative sorting decision making framework is constructed by exploring the outranking relations between alternatives and characteristic actions under multiple criteria and implementing the outranking aggregation and exploitation. Finally, this framework is demonstrated using an illustrative example with result analyses and comparison discussions. PubDate: 2020-12-01 DOI: 10.1007/s10700-020-09330-z

Abstract: Selecting financial products is one of the most fundamental investment activities to both individuals and companies, and therefore it is very important to establish an efficient and practical method for financial products selection. To address the challenge of the complicated decision-making environment and decision makers’ expression habits for the selection of financial products, this paper develops the incomplete additive probabilistic linguistic preference relation to depict decision makers’ preferences. Considering that, when decision makers express their opinions using probabilistic linguistic preference relation, it is possible that the sum of the value of the probability information is more than 1, this paper also extends the concepts of probabilistic linguistic term set, additive probabilistic linguistic preference relation and incomplete additive probabilistic linguistic preference relation to improve and ensure their practicability. Moreover, an “inverse prospect theory-based” algorithm is proposed to choose proper financial products. The algorithm processes the original incomplete additive probabilistic linguistic preference relation by using the inverse functions of the prospect theory at first. Then, a priority weights deriving model is established based on the extended concepts. Finally, the numerical case and analysis is presented as the evidences to the conclusion that the proposed algorithm is practical and robust. PubDate: 2020-11-05 DOI: 10.1007/s10700-020-09348-3

Abstract: Quality function deployment (QFD) is a customer-oriented product/service design tool with diversified team members reaching a consensus in developing a new or an improved product/service to maximize customer satisfaction. Determining the relative importance ratings (RIRs) of customer requirements (CRs) is actually a multiple attribute decision making (MADM) problem, which is regarded as an essential step in QFD application. Although many decision making approaches have been developed to rate the CRs, few of them refer to derive the weights of evaluators, and there is paucity of literature which combines the probabilistic linguistic term sets (PLTSs) with QFD methodology. Therefore, PLTSs are introduced in our study, to simultaneously reflect the hesitancy and preference degrees of evaluators. A novel MADM technique based on probabilistic linguistic preference (PLP) is explored to calculate the RIRs among CRs under the PLP environment. Concretely, some operators, such as the normalization formulas, the probabilistic linguistic expected value (PLEV) operator as well as the standard formula, are applied to weight the decision makers; a modified grey relational analysis-technique for order preference by similarity to ideal solution (GRA-TOPSIS) method called probabilistic linguistic-based GRA-TOPSIS is also proposed to rate the relative importance over CRs. Finally, application of product improvement of a turbine engine is given to see the validity and feasibility of the proposed approach. PubDate: 2020-10-26 DOI: 10.1007/s10700-020-09347-4

Abstract: The Evidential C-Means algorithm provides a global treatment of ambiguity and uncertainty in memberships when partitioning attribute data, but still requires the number of clusters to be fixed as a priori, like most existing clustering methods do. However, the users usually do not know the exact number of clusters in advance, particularly in practical engineering. To relax this requirement, this paper proposes an Evidential Evolving C-Means (E2CM) clustering method in the framework of evolutionary computation: cluster centers are encoded in a population of variable strings (or particles) to search the optimal number and locations of clusters simultaneously. To perform such joint optimization problem well, an artificial bee colony algorithm with variable strings and interactive evaluation mode is proposed. It will be shown that the E2CM can automatically create appropriate credal partitions by just requiring an upper bound of the cluster number rather than the exact one. More interestingly, there are no restrictions on this upper bound from the theoretic point of view. Some numerical experiments and a practical application in thermal power engineering validate our conclusions. PubDate: 2020-10-06 DOI: 10.1007/s10700-020-09344-7