The transformation method for the simulation and analysis of systems with uncertain parameters

In this paper, the transformation method is introduced as a powerful approach for both the simulation and the analysis of systems with uncertain model parameters. Based on the concept of α-cuts, the method represents a special implementation of fuzzy arithmetic that avoids the well-known effect of overestimation which usually arises when fuzzy arithmetic is reduced to interval computation. Systems with uncertain model parameters can thus be simulated without any artificial widening of the simulation results. As a by-product of the implementation scheme, the transformation method also provides a measure of influence to quantitatively analyze the uncertain system with respect to the effect of each uncertain model parameter on the overall uncertainty of the model output. By this, a special kind of sensitivity analysis can be defined on the basis of fuzzy arithmetic. Finally, to show the efficiency of the transformation method, the method is applied to the simulation and analysis of a model for the friction interface between the sliding surfaces of a bolted joint connection.     

A dual-interval vertex analysis method and its application to environmental decision making under uncertainty

In this study, a dual-interval vertex analysis (DIVA) method is developed, through incorporating the vertex method within an interval-parameter programming framework. The developed DIVA method can tackle uncertainties presented as dual intervals that exist in the objective function and the left- and right-hand sides of the modeling constraints. An interactive algorithm and a vertex analysis approach are proposed for solving the DIVA model. Solutions under an associated α-cut level can be generated by solving a series of deterministic submodels. They can help quantify relationships between the objective function value and the membership grade, which is meaningful for supporting in-depth analyses of tradeoffs between environmental and economic objectives as well as those between system optimality and reliability. A management problem in terms of regional air pollution control is studied to illustrate applicability of the proposed approach. The results indicate that useful solutions for planning the air quality management practices have been generated. They can help decision makers to identify desired pollution-abatement strategies with minimized costs and maximized environmental efficiencies.     

Exact bounds for the sensitivity analysis of structures with uncertain-but-bounded parameters

Based on interval mathematical theory, the interval analysis method for the sensitivity analysis of the structure is advanced in this paper. The interval analysis method deals with the upper and lower bounds on eigenvalues of structures with uncertain-but-bounded (or interval) parameters. The stiffness matrix and the mass matrix of the structure, whose elements have the initial errors, are unknown except for the fact that they belong to given bounded matrix sets. The set of possible matrices can be described by the interval matrix. In terms of structural parameters, the stiffness matrix and the mass matrix take the non-negative decomposition. By means of interval extension, the generalized interval eigenvalue problem of structures with uncertain-but-bounded parameters can be divided into two generalized eigenvalue problems of a pair of real symmetric matrix pair by the real analysis method. Unlike normal sensitivity analysis method, the interval analysis method obtains informations on the response of structures with structural parameters (or design variables) changing and without any partial differential operation. Low computational effort and wide application rang are the characteristic of the proposed method. Two illustrative numerical examples illustrate the efficiency of the interval analysis.     

Non-probabilistic convex models and interval analysis method for dynamic response of a beam with bounded uncertainty

This paper is concerned with the comparison of two non-probabilistic set-theoretical models for dynamic response measures of an infinitely long beam. The beam is on an uncertain foundation and subjected to a moving force with constant speed. The steady state vibration is analyzed with finite element method. The dynamic responses of the beam are approximated to the first-order respect of the uncertainty variables. As a rule, in convex models and interval analysis, the uncertainties are considered to be unknown, but they give out their allowable vector space. Comparing the convex models with interval analysis in mathematical proofs and numerical calculations, it’s shows that under the condition of transform an interval vector to an outer enclosed ellipsoid, the dynamic response of the infinitely long beam predicted by interval analysis is smaller than that by convex models; under the condition of transform a hyperellipsoid to an outer enclosed interval vector, the dynamic response of the infinitely long beam calculated by convex models is smaller than that by interval analysis method.     

Decoupling correlated and uncorrelated parametric uncertainty contributions for nonlinear models

For models with correlated parameters, the amount of uncertainty (generally measured by variance) in a model output contributed by a specific parameter encompasses two components: (1) the uncertainty contributed by the variations (used to represent uncertainty in the parameter) correlated with other parameters; and (2) the uncertainty contributed by the variations unique to the parameter of interest (i.e., uncorrelated variations or variations that cannot be explained by any other parameters in the model). A regression-based method has been proposed previously by Xu and Gertner (2008) [1] to decouple the correlated and uncorrelated contributions to uncertainties in model outputs by each parameter for linear models. Based on a modified version of the popular Fourier Amplitude Sensitivity Test (FAST), this paper develops a general approach for the quantification of the correlated and uncorrelated parametric uncertainty contributions in linear, nonlinear and non-monotonic models with linear or nonlinear dependence among parameters. The decoupling of correlated and uncorrelated contributions can help us determine if the uncertainty contributed by a specific parameter results from the uncertainty in itself or from its correlations with other parameters. Thus, this decoupling can be very useful in improving the understanding our modeled systems.     

A new method for ranking fuzzy numbers and its application to group decision making

In this paper, a new method for comparing fuzzy numbers based on a fuzzy probabilistic preference relation is introduced. The ranking order of fuzzy numbers with the weighted confidence level is derived from the pairwise comparison matrix based on 0.5-transitivity of the fuzzy probabilistic preference relation. The main difference between the proposed method and existing ones is that the comparison result between two fuzzy numbers is expressed as a fuzzy set instead of a crisp one. As such, the ranking order of n fuzzy numbers provides more information on the uncertainty level of the comparison. Illustrated by comparative examples, the proposed method overcomes certain unreasonable (due to the violation of the inequality properties) and indiscriminative problems exhibited by some existing methods. More importantly, the proposed method is able to provide decision makers with the probability of making errors when a crisp ranking order is obtained. The proposed method is also able to provide a probability-based explanation for conflicts among the comparison results provided by some existing methods using a proper ranking order, which ensures that ties of alternatives can be broken.     

A feasible implementation procedure for interval analysis method from measurement data

An uncertain quantification and propagation procedure via interval analysis is proposed to deal with the uncertain structural problems in the case of the small sample measurement data in this study. By virtue of the construction of a membership function, a finite number of sample data on uncertain structural parameters are processed, and the effective interval estimation on uncertain parameters can be obtained. Moreover, uncertainty propagation based on interval analysis is performed to obtain the structural responses interval according to the quantified results of the uncertain structural parameters. The proposed method can decrease the demanding on the sample number of measurement data in comparison with the classical probabilistic method. For instance, the former only need several to tens of sample data, whereas the latter usually need several tens to several hundreds of them. The numerical examples illustrate the feasibility and validity of the proposed method for non-probabilistic quantification of limited uncertain information as well as propagation analysis.     

Mehar’s method for solving fuzzy sensitivity analysis problems with LR flat fuzzy numbers

In published works on fuzzy linear programming there are only few papers dealing with stability or sensitivity analysis in fuzzy mathematical programming. To the best of our knowledge, till now there is no method in the literature to deal with the sensitivity analysis of such fuzzy linear programming problems in which all the parameters are represented by LR flat fuzzy numbers. In this paper, a new method, named as Mehar’s method, is proposed for the same. To show the advantages of proposed method over existing methods, some fuzzy sensitivity analysis problems which may or may not be solved by the existing methods are solved by using the proposed method.     

Improved probabilistic point estimation schemes for uncertainty analysis

The probabilistic point estimation (PPE) methods replace the probability distribution of the random parameters of a model with a finite number of discrete points in sample space selected in such a way to preserve limit probabilistic information of involved random parameters. Most PPE methods developed thus far match the distribution of random parameters up to the third statistical moment and, in general, could provide reasonable accurate estimation only for the first two statistical moments of model output. This study proposes two optimization-based point selection schemes for the PPE methods to enhance the accuracy of higher-order statistical moments estimation for model output. Several test models of varying degrees of complexity and nonlinearity are used to examine the performance of the proposed point selection schemes. The results indicate that the proposed point selection schemes provide significantly more accurate estimation of model output uncertainty features than the existing schemes.     

A new boundary-type scheme for sensitivity analysis using Trefftz formulation

This paper presents a new boundary-type scheme for a sensitivity analysis of the two-dimensional potential problem by using the Trefftz formulation.

Since the Trefftz method is the boundary-type solution procedure, input data generation is easier than the domain-type solution procedure. Moreover, the physical quantities are expressed by the regular equations, their sensitivities, which is derived from the direct differentiation of the original quantaties, are also regular. Therefore, they can be calculated more easily than the ordinary boundary element method using the singular boundary integral equation. The present schemes are applied to simple numerical examples in order to confirm the validity of the present formulation.     

Hybrid fuzzy least-squares regression analysis in claims reserving with geometric separation method

Claims reserving is obviously necessary for representing future obligations of an insurance company and selection of an accurate method is a major component of the overall claims reserving process. However, the wide range of unquantifiable factors which increase the uncertainty should be considered when using any method to estimate the amount of outstanding claims based on past data. Unlike traditional methods in claims analysis, fuzzy set approaches can tolerate imprecision and uncertainty without loss of performance and effectiveness. In this paper, hybrid fuzzy least-squares regression, which is proposed by Chang (2001), is used to predict future claim costs by utilizing the concept of a geometric separation method. We use probabilistic confidence limits for designing triangular fuzzy numbers. Thus, it allows us to reflect variability measures contained in a data set in the prediction of future claim costs. We also propose weighted functions of fuzzy numbers as a defuzzification procedure in order to transform estimated fuzzy claim costs into a crisp real equivalent.     

Modeling uncertainty in multi-criteria decision analysis

his paper provides a review of multiple criteria decision analysis (MCDA) for cases where attribute evaluations are uncertain. The main aim is to identify different tools which can be used to represent uncertain evaluations, and to broadly survey the available decision models that can be used to support uncertain decision making. The review includes models using probabilities or probability-like quantities; explicit risk measures such as quantiles and variances; fuzzy numbers, and scenarios. The practical assessment of uncertain outcomes and preferences associated with these outcomes is also discussed.     

The differential perturbative method applied to the sensitivity analysis for waterhammer problems in hydraulic networks

The differential perturbative method was applied to the sensitivity analysis for waterhammer problems in hydraulic networks. Starting from the classical waterhammer equations in a single-phase liquid with friction (the direct problem) the state vector comprising the piezometric head and the velocity was defined. Applying the differential method the adjoint operator, the adjoint equations with the general form of their boundary conditions, and the general form of the bilinear concomitant were calculated for a single pipe. Considering that any hydraulic network can be built by connecting different components (reservoirs, valves, pumps, tees, etc.) through pipes, the adjoint relationships for any component, as well as the final contribution to the bilinear concomitant, were calculated. Moreover, an analogy was established in which transmission and reflection coefficients can be derived for any adjoint component. The importance or adjoint function was analyzed when the piezometric head or velocity at a given position and time is chosen as the response functional. In this case, it is shown that the importance function is represented by delta-functions travelling along the hydraulic network with the propagation speed. The calculation of the sensitivity coefficients takes into account the cases in which the parameters under consideration influence the initial condition. For these cases, the calculation can be performed by solving sequentially two perturbative problems: the first one is non-steady, while the second one is steady, with an appropriate selection of a weight function coming from the unsteady perturbative problem. The discretized adjoint equations and the corresponding boundary conditions were programmed and solved by using the method of characteristics. As an example, a constant-level tank connected through a pipe to a valve discharging to atmosphere was considered. The corresponding sensitivity coefficients due to the variation of different parameters by using both the differential method and the response surface generated by the computer code WHAT, solver of the direct problem, were also calculated. The results obtained with these methods show excellent agreement.     

A new defect correction method for the Navier-Stokes equations at high Reynolds numbers

In this paper, a new defect correction method for the Navier-Stokes equations is presented. With solving an artificial viscosity stabilized nonlinear problem in the defect step, and correcting the residual by linearized equations in the correction step for a few steps, this combination is particularly efficient for the Navier-Stokes equations at high Reynolds numbers. In both the defect and correction steps, we use the Oseen iterative scheme to solve the discrete nonlinear equations. Furthermore, the stability and convergence of this new method are deduced, which are better than that of the classical ones. Finally, some numerical experiments are performed to verify the theoretical predictions and show the efficiency of the new combination.     

An algorithm for a fuzzy transportation problem to select a new type of coal for a steel manufacturing unit

Manufacturing of steel involves thermal energy intensive processes with coal as the major input. Energy generated is a direct function of ash content of coal and as such it weighs very high as regards the choice of coal. In this paper, we study a multiobjective transportation problem to introduce a new type of coal in a steel manufacturing unit in India. The use of new type of coal serves three non-prioritized objectives, viz. minimization of the total freight cost, the transportation time and the ratio of ash content to the production of hot metal. It has been observed from the past data that the supply and demand points have shown fluctuations around their estimated values because of changing economic conditions. To deal with uncertainties of supply and demand parameters, we transform the past data pertaining to the amount of supply of the ith supply point and the amount of demand of the jth demand point using level (λ,ρ) interval-valued fuzzy numbers. We use a linear ranking function to defuzzify the fuzzy transportation problem. A transportation algorithm is developed to find the non-dominated solutions for the defuzzified problem. The application of the algorithm is illustrated by numerical examples constructed from the data provided by the manufacturing unit.      

Numerical simulation based on fuzzy stochastic analysis

In this paper mathematical methods for fuzzy stochastic analysis in engineering applications are presented. Fuzzy stochastic analysis maps uncertain input data in the form of fuzzy random variables onto fuzzy random result variables. The operator of the mapping can be any desired deterministic algorithm, e.g. the dynamic analysis of structures. Two different approaches for processing the fuzzy random input data are discussed. For these purposes two types of fuzzy probability distribution functions for describing fuzzy random variables are introduced. On the basis of these two types of fuzzy probability distribution functions two appropriate algorithms for fuzzy stochastic analysis are developed. Both algorithms are demonstrated and compared by way of an example.     

A fuzzy linear programming based approach for tactical supply chain planning in an uncertainty environment

This paper models supply chain (SC) uncertainties by fuzzy sets and develops a fuzzy linear programming model for tactical supply chain planning in a multi-echelon, multi-product, multi-level, multi-period supply chain network. In this approach, the demand, process and supply uncertainties are jointly considered. The aim is to centralize multi-node decisions simultaneously to achieve the best use of the available resources along the time horizon so that customer demands are met at a minimum cost. This proposal is tested by using data from a real automobile SC. The fuzzy model provides the decision maker (DM) with alternative decision plans with different degrees of satisfaction.     

A new solution method for fuzzy chance constrained programming problem

This paper deals with an optimization model, where both fuzziness and randomness occur under one roof. The concept of fuzzy random variable (FRV), mean and variance of FRV is used in the model. In particular, the methodology is developed in the presence of FRV in the constraint. The methodology is verified through numerical examples.     

A new methodology for sensitivity and stability analysis of analytic network models

In this paper we develop a methodology to study the sensitivity and the stability of models built using the Analytic Network Process. We study two types of stability: core and solution stability. The former deals with finding the region of the perturbation space in which the initial solution (i.e., the alternative that is ranked first) obtained from the ANP model remains most preferred. The latter deals with finding the regions of the perturbation space in which the solutions that were not initially most preferred (i.e., alternatives that were not ranked first) become most preferred (i.e., they are ranked first). The methodology consists of three stages: generation of the perturbation space, finding the boundaries of the regions in the perturbation space in which the different alternatives are ranked first, and finding the stability regions.     

A new methodology for crisp equivalent of fuzzy chance constrained programming problem

This paper deals with a chance constrained programming model, where both fuzziness and randomness are present in the objective function and constraints. The concept of fuzzy random variable, mean and variance of fuzzy random variable, minimum of fuzzy numbers are used in the model. The methodology is verified through a numerical example.