Search

In this paper, we use a Caputo–Hadamard uncertain fractional differential equation (UFDE) to describe the change of carbon emission rights price. Based on a binomial tree approach, a portfolio is constructed by buying options and shorting the underlying assets in a certain proportion. The pricing formulas for carbon finance European option are presented, and the algorithms are designed to formulate the price of carbon finance American option without explicit pricing formulas. Besides, a moment estimation method is used to obtain the parameter values in Caputo–Hadamard UFDE, and a method is provided to simulate the observation data of solution of Caputo–Hadamard UFDE. Finally, the specific Caputo–Hadamard UFDE with mean-reverting process is applied to simulating the change of carbon emission rights price in Chinese carbon market, and the feasibility of the proposed carbon option pricing method is illustrated by some numerical examples. [ABSTRACT FROM AUTHOR]

With the development of mathematical theory, fractional order equation is becoming a potential tool in the context of neural networks. This paper primarily concerns with the stability for systems governed by the linear fractional order uncertain difference equations, which may properly portray neural networks. First, the solutions of these linear difference equations are provided. Secondly, the definition of finite-time stability in measure for the proposed systems is introduced. Furthermore, some sufficient conditions checking for it are achieved by the property of fractional order difference and uncertainty theory. Besides, the relationship between finite-time stability almost surely and in measure is discussed. Finally, some numerical examples are analysed by employing the proposed results.

Time-delay is a universal phenomenon in control systems, which usually leads to instability and poor performance of systems. In this paper, the p th moment asymptotic stability of trivial solutions to uncertain dynamical systems with time-delays is investigated. The concept of the generalized expected value is introduced with its properties based on uncertainty theory. Sufficient conditions for ensuring the stability of uncertain delay systems are derived by Lyapunov direct method. Several illustrative examples with numerical simulations are arranged to demonstrate the effectiveness of the stability results. [ABSTRACT FROM AUTHOR]

It is well-known that uncertain random systems suffered from a competing failure process have been applied in industrial applications. Extending previous studies, we optimize system cost combined with burn-in and preventive maintenance strategies. A system in burn-in process is only subject to uncertain continuous stationary degradation process while it experiences competitive degradation and shock processes simultaneously in the period of operation. The dependence assumes that shocks can accelerate the degradation by uncertain additional degradation shifts. A reliability optimization model for uncertain random systems is proposed where the reliability index is defined as a chance measure, and the optimal burn-in parameters and maintenance observation time are derived by the genetic algorithm. Finally, we conclude with a numerical example to illustrate the application of burn-in and maintenance methods.

This paper mainly concerns with the stability of nonlinear Caputo fractional neutral stochastic differential system with variable delay. A sufficient condition is derived to ensure that the trivial solution is mean-square asymptotically stable by employing the Banach’s contraction principle, in which the boundedness of delays is not required. An illustrative example is provided to show the efficiency of our results.

The reliability of a dynamic system is not constant under uncertain random environments due to the interaction of internal and external factors. The existing researches have shown that some complex systems may suffer from dependent failure processes which arising from hard failure and soft failure. In this paper, we will study the reliability of a dynamic system where the hard failure is caused by random shocks which are driven by a compound Poisson process, and soft failure occurs when total degradation processes, including uncertain degradation process and abrupt degradation shifts caused by shocks, reach a predetermined critical value. Two types of uncertain random optimization models are proposed to improve system reliability where belief reliability index is defined by chance distribution. Then the uncertain random optimization models are transformed into their equivalent deterministic forms on the basis of α-path, and the optimal solutions may be obtained with the aid of corresponding nonlinear optimization algorithms. A numerical example about a jet pipe servo valve is put forward to illustrate established models by numerical methods. The results indicate that the optimization models are effective to the reliability of engineering systems. It is our future work to consider an interdependent competing failure model where degradation processes and shocks can accelerate each other.

A single machine scheduling problem with periodic maintenance is studied in this paper. Due to many uncertainties in reality, the processing time is recognized as an uncertain variable. The aim is to minimize the makespan at a confidence level. An uncertain chance-constrained programming model is developed to delve into the impact of uncertainties on decision variables, and an algorithm for calculating the objective function is proposed. According to the theoretical analysis, a novel method named longest shortest processing time (LSPT) rule is proposed. Subsequently, an improved genetic algorithm (GA-M) combined with LSPT rule is proposed. Numerical experiments are used to verify the feasibility of this model and algorithm.

After uncertainty theory was established, it has become a new branch of mathematics and been applied to describing the indeterministic phenomena as an uncertain dynamic system. As an important part of uncertainty theory, uncertain fractional differential equation is a good tool to model some complex dynamic systems with uncertainties. Due to the lack of corresponding information, models of parametric uncertain fractional differential equation are not always available. Therefore, the requirement for nonparametric estimation of uncertain fractional differential equation is urgent. Utilizing the thought of Legendre polynomial approximation, we propose a method to estimate the nonparametric uncertain fractional differential equations. Then, the error analysis of this method is given. After that, the numerical simulation is applied to illustrating the validity of error analysis and show the errors between the approximating functions and actual functions. The stability of this method has also been verified without highly dense observations. Finally, by means of uncertain hypothesis test and Kolmogorov–Smirnov test, we prove the superiority of uncertain fractional differential equations for some examples. • A nonparametric estimation method is proposed for uncertain fractional differential equations (UFDEs). • Error analysis of the proposed method is given and illustrate with numerical simulation. • Advantages of UFDEs are proved by making comparison with uncertain differential equations and stochastic differential equations. [ABSTRACT FROM AUTHOR]

A parallel-machine scheduling problem with preventive maintenance is studied in the paper. Because of the existence of indeterminacy phenomenon, the processing and maintenance time are assume to be uncertain variables. To address these uncertain parameters, two pessimistic value mathematical programming models are developed and the equivalence is proved. The worst-case bound of longest processing time (LPT) rule for the problem under an uncertain environment is obtained. To solve this problem effectively, an improved LPT rule is proposed according to the result of the theoretical analysis. Finally, numerical experiments show that the improved LPT outperforms the LPT in improving the quality of the optimal value and has a less relative error than that of the LPT.

In this paper, a linear quadratic (LQ) optimal control model is studied for a multistage uncertain random system. First, recurrence equations are presented for this model based on Bellman’s Principle. Second, analytical expressions are derived for solving both the optimal objective function and the optimal control strategy of the model. Finally, an example is given to illustrate our results.

In this paper, we investigate uncertain saddle point equilibrium differential games under uncertain environment. We propose an optimistic value game model and define the value function of the game by introducing the concept of non-anticipating strategy. We prove the continuity and dynamic programming property of the value function. Then we derive the uncertain Hamilton–Jacobi–Isaacs equation by the viscosity solution approach.

As it is well known, the optimal control of linear quadratic model is given in a feedback form, which is determined by the solution of a Riccati equation. However, the corresponding Riccati equation cannot be solved analytically in many cases. Even if an analytic solution can be obtained, it might be a complex time-oriented function. In this paper, we introduce an approximate model with parameter for simplifying the form of optimal control of uncertain linear quadratic model. First, we discuss an optimal control problem of uncertain linear quadratic model and deduce an analytic expression of optimal control. Then we formulate an approximate model with parameter and present a parametric optimization method for solving the optimal parameter. Finally, a production planning problem is given to illustrate the efficiency of the proposed approximate model and parametric optimization approach.

In this paper, we investigate pricing problems of geometric average Asian barrier options based on a Caputo type of uncertain fractional differential equation. Unlike traditional stock models based on stochastic/ uncertain processes, we assume that underlying assets are subject to uncertain fractional processes. First, it provides a Caputo type of uncertain fractional mean-reverting model based on fractional differential equation. Then the pricing formulas are acquired for four geometric average Asian barrier options, namely, the up-and-in call option, the down-and-in put option, the up-and-out put option and the down-and-out call option. Meanwhile, numerical simulations are devised to validate the results of the model.

We investigate in this paper a version of pure exchange competitive equilibrium under uncertain circumstances. Those uncertain factors are embedded in each agent’s preference, which is characterized by the uncertain utility function. By maximizing the expected utility of each agent, we formulate this kind of pure exchange competitive equilibrium problem into a quasi-variational inequality problem. This idea is applied in a pure exchange economy which consists of two agents and two goods. And we find the competitive equilibrium of this economy with each agent’s preference being an uncertain variable.

In this paper, problems of stability in measure and optimal control are considered for an uncertain continuous-time singular system which can be described by uncertain differential equations. Singular system is assumed to be regular and impulse-free. Under some given assumptions, the existence and uniqueness of the solution, and stability in measure of the singular system are presented. Then, by applying the principle of optimality and uncertainty theory, the equation of optimality for the optimal control model subject to the uncertain singular system is obtained. In the end of Sections 3–5, three examples are given to illustrate the results obtained.

This paper is primarily dedicated to the necessary conditions of optimal control for uncertain discrete-time systems of fractional-order governed by left Riemann-Liouville-like fractional-order nabla difference equations. First, an existence result of solutions for backward right Riemann-Liouville-like fractional-order nabla difference equations is derived from the Banach contraction mapping theorem. Besides, necessary optimality conditions of these fractional-order uncertain optimal control problems are proposed by the variational method and Lagrange multiplier technique. Finally, a special LQ optimal control problem for uncertain discrete-time systems of fractional-order is solved by necessary conditions of optimality derived.

In this paper, two generalized uncertain fractional differential equations (UFDEs) are proposed, which extend Riemann–Liouville type and Hadamard type into a uniform modality. Then the analytical solutions to several linear generalized (Caputo) UFDEs are provided. Furthermore, an existence and uniqueness theorem of solution for a generalized Caputo UFDE is proven.

Typical degradation-shock failure processes have been widely investigated in current researches, and the failures caused by their dependence are described as competitive failure processes. This paper explores competitive failure modes for uncertain random fractional systems involving degradation and shock processes. We develop a wear degradation model explicitly by employing uncertain fractional differential equations in order to demonstrate the potential heredity and memorability of a system. External shocks are then considered to follow a Poisson process. Based on the classification of shock types, three definitions of reliability index for competitive failures are presented. The reliability index formulas are derived for systems with extreme shock, cumulative shock, and δ shock using chance measures. Finally, we introduce a numerical example where the results of the reliability analysis confirm the validity of proposed reliability evaluation methods. • Present three types of reliability models in uncertain random environments • Describe the degradation process by uncertain fractional differential equations • Define the reliability index of competitive failure models by chance measure • Propose three specific reliability index formulas of the proposed models • Introduce a numerical example of jet pipe servo valves [ABSTRACT FROM AUTHOR]

This paper deals with the impulsive problem for uncertain fractional dynamical system. Firstly, the concept of uncertain fractional impulsive problem involving the Caputo derivative is introduced and the analytic solutions to several linear uncertain fractional impulsive problems are derived with the help of the Mittag-Leffler functions. Then the existence and uniqueness theorems are developed via the Kuratowski measure of noncompactness and fixed point theorems, respectively. Finally, an illustrative example is provided to explain our main results. [ABSTRACT FROM AUTHOR]

• Present an optimal control model with multiple time-delays in uncertain random environments. • A method is presented to transform the problem into an equivalent optimization problem. • Propose the recurrence equations to solve the optimization problem. • The optimal solutions of two kinds of optimization problems are obtained. Chance theory provides us a useful tool to solve an optimal control problem with indeterminacy composing of both uncertainty and randomness. Based on chance theory, this paper studies an optimal control for uncertain random singular systems with multiple time-delays. First, an uncertain random singular system with multiple time-delays is introduced, and then the corresponding optimal control problem is established. The equivalent relationship between this problem and the optimal control problem for standard uncertain random systems is derived. Then the appropriate recurrence equations are proposed according to the dynamic programming method. Furthermore, two kinds of optimal control problems are discussed. The optimal control inputs and respective optimal values of the problems are provided via the solvability of the obtained equations. Finally, a numerical example is presented to show the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: An X-valued rough variable is introduced as a measurable function from a rough space to a measurable space X. Based on the concept of X-valued rough variables, some properties of X-valued rough variables are studied. Furthermore, the concept of χ-convergence is presented, and χ-convergence of X-valued rough variables is discussed. [Copyright &y& Elsevier]

By using fixed point index approach for multivalued mappings, the existence of nonzero solutions for a class of generalized variational inequalities is studied in reflexive Banach space. One of the mappings concerned here is coercive or monotone and the other is set-contractive or upper semicontinuous.

A random fuzzy variable is a function from a possibility space to the set of random variables. In this paper, some mathematical properties of a random fuzzy variable are studied. First, some properties of chance measure of random fuzzy events are given. Then, several continuity theorems for random fuzzy sequences are investigated. Furthermore, the concept of chance distribution for a random fuzzy variable is presented, and some properties of chance distribution are obtained.

• Expected value model of LQ optimal control problem for fractional-order discrete time uncertain systems is proposed. • The proposed expected value model is indirectly solved by dynamic programming method. • A new expected value model of LQ optimal control problem for macroeconomic systems is established and solved. This paper primarily focuses on LQ optimal control problem of fractional-order discrete-time systems based on uncertainty theory that is a significant tool to model belief degree. First, an equivalent LQ problem, which is integer-order one, is presented by expanding the state variables with the unchanging control variables. This equivalent problem is then solved by dynamic programming approach. Accordingly, the solution of the proposed LQ optimal control problem reduces to solve a system of backward matrix difference equations. Finally, a numerical example is provided to show how to solve the proposed LQ optimal control problem. As an application, an LQ optimal control problem of macroeconomic system is discussed by the achieved results. [ABSTRACT FROM AUTHOR]

• Introduce a concept of Caputo-Hadamard uncertain fractional differential equation (UFDE). • Present the analytical solution to a linear Caputo-Hadamard UFDE. • Prove an existence and uniqueness theorem of solution to Caputo-Hadamard UFDE. The tool of uncertain fractional differential equations (UFDEs) is devoted to describing the behavior of complex systems with memory effects in the uncertain environment. In this paper, we mainly investigate the Caputo-Hadamard UFDEs. First, the definition of Caputo-Hadamard UFDE is proposed and the analytical solution to a linear Caputo-Hadamard UFDE is provided. Then, an existence and uniqueness theorem of solution to Caputo-Hadamard UFDE is studied. [ABSTRACT FROM AUTHOR]

Uncertain differential equation (UDE) is a type of differential equation driven by Liu process, which is used to describe uncertain dynamic systems. Since the analytic solution of UDE is difficult to obtain or the representation is complex, this paper proposes a new Adams predictor–corrector method to solve UDE. Compared with Euler Method, Runge–Kutta method, Adams method, Milne method, Adam–Simpson method and Hamming method of solving UDE, the proposed algorithm has higher accuracy. The extreme value and time integral of solution to UDE are obtained in virtue of the proposed method. Meanwhile, the convergence properties of the method are discussed. Finally, some illustrated examples are provided to demonstrate the efficiency and accuracy of the proposed technique.

Compared with investing an ordinary options, investing the power options may possibly yield greater returns. On the one hand, the power option is the best choice for those who want to maximize the leverage of the underlying market movements. On the other hand, power options can also prevent the financial market changes caused by the sharp fluctuations of the underlying assets. In this paper, we investigate the power option pricing problem in which the price of the underlying asset follows the Ornstein–Uhlenbeck type of model involving an uncertain fractional differential equation. Based on critical value criterion, the pricing formulas of European power options are derived. Finally, some numerical experiments are performed to illustrate the results.

The intuitionistic fuzzy soft set (IFSS) is one of the useful mathematical tools for uncertainty description and has many applications in real-world decision-making problems. However, the computations become more complex when these decision-making problems involve less important or redundant parameters. To solve this problem, in this paper, we study the problem of parameter reduction of IFSS based on evaluation score criteria. Initially, we developed a new approach to IFSS-based decision-making. Then using the new decision criteria, we propose three different algorithms for parameter reduction of IFSSs satisfying the different needs of decision-makers. We compare the proposed algorithms with Ghosh et al.’s algorithms in terms of different aspects. It is evident from the comparison results that the proposed algorithms are much better than Ghosh et al.’s algorithms in terms of efficiency and applicability. We also provide a comparative study among the new algorithms to decide their feasibilities in different situations. Finally, we take a university recruitment problem to verify the effectiveness of the proposed algorithms in real-life decision-making problems.

• Present first hitting time theorems for uncertain fractional differential equations. • Provide efficient algorithms for first hitting time of solutions to UFDE. • Calculate the risk index for a novel uncertain stock model. Uncertain fractional differential equation with memory and hereditary characteristics is a useful way to better model the uncertain dynamic system. Firstly, the solution of an uncertain fractional differential equation with the Caputo type is considered and uncertain distributions of their first hitting time is investigated. On the basis of the α -path, two different first hitting time theorems for uncertain distributions are proposed. Secondly, by the predictor-corrector method, the numerical method is designed. A nonlinear example is provided for validating the availability of the proposed algorithm. Then, as an application of the first hitting time, a novel uncertain risk index model is presented and a formula of risk index under our model is derived accordingly. Lastly, the numerical algorithm of risk index is designed and numerical calculations for the risk index are illustrated with regard to different parameters. [ABSTRACT FROM AUTHOR]

Abstract A 6-dimension dynamical Schistosomiasis model incorporating 5 time delays is established in order to investigate the effects of time delays on the transmission mechanisms of Schistosomiasis. The stability behaviors at the endemic equilibrium are presented under different cases, which relate to the threshold value for disease breakout. The following results and available control strategies can be obtained from theoretical analyses and numerical simulations. (1) The incubation of egg hatching can postpone the outbreak of Schistosomiasis. Therefore it can be used as a new method for the control of Schistosomiasis spread that prolonging the incubation of egg hatching. (2) The time delays in miracidia and cercaria can extend the disease transmission cycle. Thus lengthening the incubation periods in miracidia and cercaria can be beneficial to Schistosomiasis prevention at the beginning. (3) The maturation periods of schistosome and sporocyst could give rise to Hopf bifurcations and periodic orbits, respectively, and the basic reproductive number decreases with respect to the maturation of sporocyst. Therefore, it is a reasonable strategy to control the spread of Schistosomiasis that prolonging the maturation duration of the sporocyst in snails. The consequences obtained motivate us that we can choose the right time to do the control laws, such as in winter (when lower temperature). [ABSTRACT FROM AUTHOR]

A 6-dimension dynamical schistosomiasis model incorporating five time delays is established in this paper. Two equilibrium points: a disease free equilibrium and an endemic equilibrium, are calculated respectively. The stability behaviors at the disease free equilibrium are analysed. Both analytical and numerical results are presented that prepatent periods in infection can affect the schistosomiasis transmission significantly. Thus, two effective measures on schistosomiasis prevention and control are obtained: lengthening the prepatent period in susceptible snails, and prolonging the incubation periods in miracidia and cercaria by temperature control or drug restraint. And then, numerical simulations are given to illustrate the validity and effectiveness of the model. At last a discussion is provided about our results and further work. [ABSTRACT FROM AUTHOR]