Selection of efficient portfolios–probabilistic and fuzzy approach,comparative study

This paper presents a new method for selection of efficient portfolios in a situation where parameters in the calculation of effectiveness are expressed in form of interactive fuzzy numbers and probability distribution. Fuzzy simulations are used for performing arithmetic operations on interactive fuzzy numbers. The process of selecting investment projects takes into account statistical and economic dependencies of projects. The paper contains comparison of results obtained in selection of efficient portfolios of investment projects in a situation where parameters in calculation of effectiveness were expressed in form of interactive fuzzy numbers or probability distributions.     

Joint Propagation and Exploitation of Probabilistic and Possibilistic Information in Risk Assessment

Random variability and imprecision are two distinct facets of the uncertainty affecting parameters that influence the assessment of risk. While random variability can be represented by probability distribution functions, imprecision (or partial ignorance) is better accounted for by possibility distributions (or families of probability distributions). Because practical situations of risk computation often involve both types of uncertainty, methods are needed to combine these two modes of uncertainty representation in the propagation step. A hybrid method is presented here, which jointly propagates probabilistic and possibilistic uncertainty. It produces results in the form of a random fuzzy interval. This paper focuses on how to properly summarize this kind of information; and how to address questions pertaining to the potential violation of some tolerance threshold. While exploitation procedures proposed previously entertain a confusion between variability and imprecision, thus yielding overly conservative results, a new approach is proposed, based on the theory of evidence, and is illustrated using synthetic examples.     

Possibility theory and statistical reasoning

Numerical possibility distributions can encode special convex families of probability measures. The connection between possibility theory and probability theory is potentially fruitful in the scope of statistical reasoning when uncertainty due to variability of observations should be distinguished from uncertainty due to incomplete information. This paper proposes an overview of numerical possibility theory. Its aim is to show that some notions in statistics are naturally interpreted in the language of this theory. First, probabilistic inequalites (like Chebychev's) offer a natural setting for devising possibility distributions from poor probabilistic information. Moreover, likelihood functions obey the laws of possibility theory when no prior probability is available. Possibility distributions also generalize the notion of confidence or prediction intervals, shedding some light on the role of the mode of asymmetric probability densities in the derivation of maximally informative interval substitutes of probabilistic information. Finally, the simulation of fuzzy sets comes down to selecting a probabilistic representation of a possibility distribution, which coincides with the Shapley value of the corresponding consonant capacity. This selection process is in agreement with Laplace indifference principle and is closely connected with the mean interval of a fuzzy interval. It sheds light on the “defuzzification” process in fuzzy set theory and provides a natural definition of a subjective possibility distribution that sticks to the Bayesian framework of exchangeable bets. Potential applications to risk assessment are pointed out.     

A fuzzy-probabilistic earthquake risk assessment system

This paper presents an outline of a risk assessment system for evaluating the expected damage of structures and the consequent financial losses and casualties due to a likely earthquake under elevated uncertain conditions; namely where neither the statistical data nor the seismological and engineering knowledge required for such evaluations are sufficient. In such cases, we should consider extra dimensions of uncertainty, in addition to probability that is usually sufficient for expressing the risk of losses and casualties due to an earthquake where such knowledge and data is available. In the present paper, the uncertainties caused by the insufficient knowledge about the interdependency of various parameters have been considered by means of fuzzy relations. Moreover, the uncertainties in eliciting the likelihood of the seismic hazard have been expressed by fuzzy probability in the form of possibility-probability distributions (PPDs). In other words, fuzzy set theory is employed to complement the standard probability theory with a second dimension of uncertainty. By composition of the fuzzy probability of the seismic hazard and fuzzy vulnerability relation of target structure, the fuzzy probability of damage can be derived. The proposed approach has also been compared with an alternative approach for obtaining a PPD of the hazard. As a case study, the risk assessment system has been tested on a sample structure in the Istanbul metropolitan area.     

基于模糊集鲁棒概率不等式的风电斜坡估计方法

从不确定性量化的角度将风电出力的斜坡事件发生概率转化为一个数据驱动的鲁棒概率不等式,该不等式提供一个随机变量的概率,且其概率分布属于给定的多面体.为了解决风电输出的分布不确定性问题,考虑由可用数据构造的模糊集合中的候选分布集合,其最低要求是要包括预测值和平均绝对偏差.基于矩的模糊集由所有概率分布组成,共享相同的平均和平...     

Diagnostic reasoning and medical decision-making with fuzzy influence diagrams

Influence diagrams have been widely used as knowledge bases in medical informatics and many applied domains. In conventional influence diagrams, the numerical models of uncertainty are probability distributions associated with chance nodes and value tables for value nodes. However, when incomplete knowledge or linguistic vagueness is involved in the reasoning systems, the suitability of probability distributions is questioned. This study intends to propose an alternative numerical model for influence diagrams, possibility distributions, which extend influence diagrams into fuzzy influence diagrams. In fuzzy influence diagrams, each chance node and value node is associated with a possibility distribution which expresses the uncertain features of the node. This study also develops a simulation algorithm and a fuzzy programming model for diagnosis and optimal decision in medical settings.     

基于直觉模糊可能性分布的三支决策模型的研究

直觉模糊集理论和可能性理论的融合是不确定问题领域的一个研究热点。文中提出了一种基于直觉模糊可能性分布的直觉模糊可能性测度(Intuitionistic Fuzzy Probability Measurement,IFPM),并在此基础上构建了三支决策模型。首先,定义了直觉模糊决策空间及该空间上的直觉模糊可能性分布,并对其性质进行了证明,给出了论域对象的隶属度和非隶属度可能性均值的计算方法。然后,讨论了论域对象的隶属度和非隶属度可能性均值与决策阈值的关系,分析了它们之间的概率分布情况。根据概率分布－可能性分布的转换关系,给出决策规则和三支决策模型,提出了一种基于直觉模糊可能性分布的IFPM决策风险计算方法。最后,考虑论域中对象的增减变化引起的IFPM变化,给出对应公式并对动态决策过程进行分析,同时通过实例验证了该模型的有效性。     

Selection of appropriate defuzzification methods: Application to the assessment of dam performance

The performance assessment of dams is of major importance for optimizing maintenance management. A methodology is provided here to assess the risk of dam failure using data collected during in-situ inspections and from the design and follow-up files. This assessment takes into account uncertainty associated with the data and the assessment process. The result of this assessment is introduced in a fuzzy frame. This paper presents the approach taken to choose defuzzification methods that allow extracting the information necessary from this possibility distribution so that dam experts can both rank dam maintenance actions and communicate the results of this assessment.This paper first presents the process used to assess dam performance after which it presents the defuzzification methods available. A sensitivity analysis is then performed to select the methods most relevant for our case study. The last part presents the development of a tool useful for the application of defuzzification methods to our problematic and describes a real case study in which the selected defuzzification methods are used.     

Optimizing an equilibrium supply chain network design problem by an improved hybrid biogeography based optimization algorithm

This paper presents a new equilibrium optimization method for supply chain network design (SCND) problem under uncertainty, where the uncertain transportation costs and customer demands are characterized by both probability and possibility distributions. We introduce cost risk level constraint and joint service level constraint in the proposed optimization model. When the random parameters follow normal distributions, we reduce the risk level constraint and the joint service level constraint into their equivalent credibility constraints. Furthermore, we employ a sequence of discrete possibility distributions to approximate continuous possibility distributions. To enhance solution efficiency, we introduce the dominance set and efficient valid inequalities into deterministic mixed-integer programming (MIP) model, and preprocess the valid inequalities to obtain a simplified nonlinear programming model. After that, a hybrid biogeography based optimization (BBO) algorithm incorporating new solution presentation and local search operations is designed to solve the simplified optimization model. Finally, we conduct some numerical experiments via an application example to demonstrate the effectiveness of the designed hybrid BBO.     

Modeling data envelopment analysis by chance method in hybrid uncertain environments

This article first presents several formulas of chance distributions for trapezoidal fuzzy random variables and their functions, then develops a new class of chance model (C-model for short) about data envelopment analysis (DEA) in fuzzy random environments, in which the inputs and outputs are assumed to be characterized by fuzzy random variables with known possibility and probability distributions. Since the objective and constraint functions contain the chance of fuzzy random events, for general fuzzy random inputs and outputs, we suggest an approximation method to compute the chance. When the inputs and outputs are mutually independent trapezoidal fuzzy random variables, we can turn the chance constraints and the chance objective into their equivalent stochastic ones by applying the established formulas for the chance distributions. In the case when the inputs and the outputs are mutually independent trapezoidal fuzzy random vectors, the proposed C-model can be transformed to its equivalent stochastic programming one, in which the objective and the constraint functions include a number of standard normal distribution functions. To solve such an equivalent stochastic programming, we design a hybrid algorithm by integrating Monte Carlo (MC) simulation and genetic algorithm (GA), in which MC simulation is used to calculate standard normal distribution functions, and GA is used to solve the optimization problems. Finally, one numerical example is presented to demonstrate the proposed modeling idea and the efficiency in the proposed model.     

Numerical accuracy and efficiency in the propagation of epistemic and aleatory uncertainties

The need to differentiate between epistemic and aleatory uncertainties is now well admitted by the risk analysis community. One way to do so is to model aleatory uncertainty by classical probability distributions and epistemic uncertainty by means of possibility distributions, and then propagate them by their respective calculus. The result of this propagation is a random fuzzy variable. When dealing with complex models, the computational cost of such a propagation quickly becomes too high. In this paper, we propose a numerical approach, the Random/Fuzzy (RaFu) method, whose aim is to determine an optimal numerical strategy so that computational costs are reduced to their minimum, using the theoretical frameworks mentioned above. We also give some means to take account of the resulting numerical error. The benefits of the RaFu method are shown by comparing it to previously proposed methods.     

On weighted lower and upper possibilistic means and variances of fuzzy numbers and its application in decision

In this paper, we define the weighted lower and upper possibilistic variances and covariances of fuzzy numbers. We also obtain their many properties similar to variance and covariance in probability theory. On the basis of the weighted lower and upper possibilistic means and variances, we present two new possibilistic portfolio selection models with tolerated risk level and holdings of assets constraints. The conventional probabilistic mean–variance model can be transformed to a linear programming problem under possibility distributions. Finally, an estimation method of possibility distribution is offered and a real example for portfolio selection problem is given to illustrate the usability of the approach and the effectiveness of our methods.     

Construction of families of probability boxes and corresponding membership functions at different fractiles

Uncertainty comes in many forms in the real world and is an unavoidable component of human life. Generally, two types of uncertainties arise, namely, aleatory and epistemic uncertainty. Probability is a well established mathematical tool to handle aleatory uncertainty and fuzzy set theory is a tool to handle epistemic uncertainty. However, in certain situations, parameters of probability distributions may be tainted with epistemic uncertainty; and so, representation of parameters of probability distributions may be treated as fuzzy numbers (may be of different shapes). A probability box (P‐box) can be constructed when parameters are not precisely known. In this paper, an attempt has been made to construct families of P‐boxes when parameters of probability distributions are bell shaped or normal fuzzy numbers; and from these families of P‐boxes, membership functions are generated at different fractiles for different alpha levels.     

基于仿真的费用优化与风险分析软件系统

Arena仿真软件环境和Excel2000统计软件环境下，设计开发了一个针对特定随机型活动网络计划的进度／费用风险分析以及费用分析、均衡优化的集成化软件系统。该系统能实现特定随机型网络计划模型中各个活动时问和费用及其概率分布参数的输入和修改、仿真建模及运行、仿真结果动态可视化输出、网络计划的进度／费用仿真及其风险概率估计、代表性样本点的选择、网络计划各活动Weibull分布的参数估计、整个工期多峰Welbull分布的生成及投资强度分布的均衡优化，可埔于辅助武器装备系统研制项目及其他大型工程项目的风险分析与投资管理。     

Sensitivity analysis of structural response uncertainty propagation using evidence theory

Sensitivity analysis for the quantified uncertainty in evidence theory is developed. In reliability quantification, classical probabilistic analysis has been a popular approach in many engineering disciplines. However, when we cannot obtain sufficient data to construct probability distributions in a large-complex system, the classical probability methodology may not be appropriate to quantify the uncertainty. Evidence theory, also called Dempster–Shafer Theory, has the potential to quantify aleatory (random) and epistemic (subjective) uncertainties because it can directly handle insufficient data and incomplete knowledge situations. In this paper, interval information is assumed for the best representation of imprecise information, and the sensitivity analysis of plausibility in evidence theory is analytically derived with respect to expert opinions and structural parameters. The results from the sensitivity analysis are expected to be very useful in finding the major contributors for quantified uncertainty and also in redesigning the structural system for risk minimization.     

Evaluating methods of investment project and optimizing models of portfolio selection in fuzzy uncertainty

This paper deals with the problems of both project valuation and portfolio selection under the assumption that the investment capitals and the net cash flows of the projects are fuzzy variables. Using the credibilistic expected value and the credibilistic lower semivariance of fuzzy variables, this paper proposes both the credibilistic return index and the credibilistic risk index, which are measures of investment return and investment risk with annuity form for evaluating single project. Moreover, a composite risk-return index for selecting the optimal investment strategy is also presented. Then, we set up a general project portfolio optimization model with fuzzy returns and two specific models: triangle and interval fuzzy returns. Furthermore, we provide two algorithms: the improved heuristic rules based on genetic algorithm and the traversal algorithm. Finally, two numerical examples are presented to illustrate the efficiency and the effectiveness of these proposed optimization methods.     

Investment project valuation based on a fuzzy binomial approach

The typical approaches to project valuation are based on discounted cash flows (DCF) analysis which provides measures like net present value (NPV) and internal rate of return (IRR). DCF-based approaches exhibit two major pitfalls. One is that DCF parameters such as cash flows cannot be estimated precisely in an uncertain decision making environment. The other one is that the values of managerial flexibilities in investment projects cannot be exactly revealed through DCF analysis. Both of them would have significant influence on strategic investment projects valuation. This paper proposes a fuzzy binomial approach that can be used in project valuation under uncertainty. The proposed approach also reveals the value of flexibilities embedded in the project. Furthermore, this paper provides a method to compute the mean value of a project’s fuzzy NPV. The project’s fuzzy NPV is characterized with right-skewed possibilistic distribution because these flexibilities retain the upside potential of profit but limit the downside risk of loss. Finally, this paper discusses the value of multiple options in a project.     

IT项目进度计划风险量化预测方法

针对IT项目进度基线定义质量普遍偏低的问题,提出了一种资源约束条件下基于蒙特卡洛模拟的进度风险量化评估方法．首先,考虑资源对进度的限制问题,提出资源约束条件下的活动历时抽样方法;其次,提出在活动历时随机变化情况下,项目关键路径动态识别方法,研究项目工期概率分布问题．最后,通过实例表明,资源约束条件下的蒙特卡洛模拟方法,能够有效地评估IT项目中的进度风险,辅助项目决策．     

Distributionally robust fuzzy project portfolio optimization problem with interactive returns

Effective project selection and staff assignment strategies directly impact organizational profitability. Based on critical value optimization criterion, this paper discusses how uncertainty and interaction impact the project portfolio return and staff allocation. Since the exact possibility distributions of uncertain parameters in practical project portfolio problems are often unavailable, we adopt variable parametric possibility distributions to characterize uncertain model parameters. Furthermore, this paper develops a novel parametric credibilistic optimization method for project portfolio selection problem. According to the structural characteristics of variable parametric possibility distributions, we derive the equivalent analytical expressions of credibility constraints, and turn the original credibilistic project portfolio model into its equivalent nonlinear mixed-integer programming models. To show the advantages of the proposed parametric credibilistic optimization method, some numerical experiments are conducted by setting various values of distribution parameters. The computational results support our arguments by comparing with the optimization method under fixed possibility distributions.