Use of Intervals and Possibility Distributions in Economic Analysis

Traditional approaches to capital budgeting are based on the premise that probability theory is necessary and sufficient to deal with the uncertainty and imprecision which underlie the estimates of required parameters. This paper argues that, in many circumstances, this premise is invalid since the principal sources of uncertainty are often non-random in nature and relate to the fuzziness rather than the frequency of data. To capture and quantify correctly the underlying uncertainty present in non-statistical situations, this paper suggests two alternative representations: interval analysis and possibility distributions. The use of these representations in economic analysis is discussed, and their application is illustrated through numerical examples.     

Ambiguity aversion and ambiguity content in decision making under uncertainty

The Subjectively Weighted Linear Utility (SWLU) model for decision making under uncertainty can accommodate non-neutral attitudes toward ambiguity. We first characterize ambiguity aversion in terms of the SWLU model parameters. In addition, we show that ambiguity content may reasonably be regarded as residing in the decision maker's subjective probability distribution of induced utility. In particular, (a) a special kind of mean preserving spread of the induced utility distribution will always increase ambiguity content, and (b) utility distributions which are more shiftable by new information have higher ambiguity content.     

Combined analysis of unique and repetitive events in quantitative risk assessment

For risk assessment to be a relevant tool in the study of any type of system or activity, it needs to be based on a framework that allows for jointly analyzing both unique and repetitive events. Separately, unique events may be handled by predictive probability assignments on the events, and repetitive events with unknown/uncertain frequencies are typically handled by the probability of frequency (or Bayesian) approach. Regardless of the nature of the events involved, there may be a problem with imprecision in the probability assignments. Several uncertainty representations with the interpretation of lower and upper probability have been developed for reflecting such imprecision. In particular, several methods exist for jointly propagating precise and imprecise probabilistic input in the probability of frequency setting. In the present position paper we outline a framework for the combined analysis of unique and repetitive events in quantitative risk assessment using both precise and imprecise probability. In particular, we extend an existing method for jointly propagating probabilistic and possibilistic input by relaxing the assumption that all events involved have frequentist probabilities; instead we assume that frequentist probabilities may be introduced for some but not all events involved, i.e. some events are assumed to be unique and require predictive – possibly imprecise – probabilistic assignments, i.e. subjective probability assignments on the unique events without introducing underlying frequentist probabilities for these. A numerical example related to environmental risk assessment of the drilling of an oil well is included to illustrate the application of the resulting method.     

Sensory evaluation based on linguistic decision analysis

The evaluation processes are widely used for quality inspection, design, marketing exploitation and other fields in industrial companies. In many of these fields the items, products, designs, etc., are evaluated according to the knowledge acquired via human senses (sight, taste, touch, smell and hearing), in such cases, we talk about sensory evaluation, in it an important problem arises as it is the modelling and management of uncertain knowledge in the evaluation process, because the information acquired by our senses throughout human perceptions always involves uncertainty, vagueness and imprecision.The decision analysis techniques have been utilized in many evaluation processes, hence this paper proposes and shows the application of the linguistic decision analysis to sensory evaluation and its advantages, particularly based on the linguistic 2-tuple representation model, in order to model and manage consistently the uncertainty and vagueness of the information in this type of problems.     

Handling uncertainty in route choice models: From probabilistic to possibilistic approaches

In order to design effective advanced traffic information systems (ATIS) suitable mathematical models have to be defined to simulate the effects of information on users route choice behaviour and then to incorporate it into traffic assignment models to estimate how traffic demand loads the roads network.To face this problem it is necessary to deal with uncertainty that plays a crucial role in the users decision-making processes.To this purpose this paper first analyses how uncertainty affects users’ route choice process and how traffic assignment models may take it into account.In literature route choice behaviour modelling is widely solved within the random utility theory framework but, we show in this paper that such an approach only considers one type of uncertainty. More precisely, the consideration of randomness of traffic by drivers is, for example, hardly ever represented in classical models in spite of its importance in the management of information by drivers.Starting from the presented analysis a new route choice model is also proposed to represent explicitly the uncertainty lying in users’ route choice behaviour. It is based on recent developments in possibility theory which is an alternate way to probability theory in order to represent or measure uncertainty.     

Modeling fuzzy information in UML class diagrams and object-oriented database models

Conceptual data modeling has become essential for non-traditional application areas. Some conceptual data models have been proposed as tools for database design and object-oriented database modeling. Information in real-world applications is often vague or ambiguous. Currently, a little research is underway on modeling the imprecision and uncertainty in conceptual data modeling and the conceptual design of fuzzy databases. The unified modeling language (UML) is a set of object-oriented modeling notations and a standard of the object management group (OMG) with applications to many areas of software engineering and knowledge engineering, increasingly including data modeling. This paper introduces different levels of fuzziness into the class of UML and presents the corresponding graphical representations, with the result that UML class diagrams may model fuzzy information. The fuzzy UML data model is also formally mapped into the fuzzy object-oriented database model.     

Constructively simple estimating: a project management example

A ‘constructively simple’ approach to estimating uses a decision support modelling paradigm based on project risk management and operational research concepts. It employs probability models selected from a set of alternative stochastic models of uncertainty with a view to maximising the insight provided, given an appropriate level of complexity. It addresses issues that include the joint use of subjective and objective probabilities, subjectivity of model data and structure, bias, data acquisition costs, the importance of getting an estimate right, optimising the estimating processes involved as a whole in approximate but robust terms, and differences in interpretation of what this means to estimators and users of estimates. Specific applications are necessarily context-specific to some extent, but the underlying ideas are of general applicability. This paper uses a simple example involving estimating the uncertain duration of a project activity to illustrate what is involved.     

A method for processing the unreliable expert judgments about parameters of probability distributions

A method for combining two types of judgments about an object analyzed, which are elicited from experts, is considered in the paper. It is assumed that the probability distribution of a random variable is known, but its parameters may be determined by experts. The method is based on the use of the imprecise probability theory and allows us to take into account the quality of expert judgments, heterogeneity and imprecision of information supplied by experts. An approach for computing “cautious” expert beliefs under condition that the experts are unknown is studied. Numerical examples illustrate the proposed method.     

Uncertainty analysis of transport of water and pesticide in an unsaturated layered soil profile using fuzzy set theory

Incomplete information is notoriously common in planning soil and groundwater remediation. For making decisions groundwater flow and transport models are commonly used. However, uncertainty in prediction arises due to imprecise information on flow and transport parameters like saturated/unsaturated hydraulic conductivity, water retention curve parameters, precipitation and evapo-transpiration rates as well as factors governing the fate of pollutant in soil like dispersion, diffusion, degradation and chemical transformation. Different methods exist for quantifying uncertainty, e.g. first and second order Taylor’s Series and Monte-Carlo method. In this paper, a methodology based on fuzzy set theory is presented to express imprecision of input data, in terms of fuzzy number, to quantify the uncertainty in prediction. The application of the fuzzy set theory is demonstrated through pesticide (endosulfan) transport in an unsaturated layered soil profile. The governing partial differential equation along with fuzzy inputs, results in a non-linear optimization problem. The solution gives complete membership functions for flow (suction head) and pesticide concentration in soil column.     

Influence of measurement data metrics on the identification of Interval Fields for the representation of spatial variability in finite element models

Due to the exponential growth in computing power, numerical modelling techniques method have gained an increasing amount of interest for engineering and design applications. Nowadays, the deterministic finite element (FE) method, an efficient tool to accurately solve the Partial Differential Equations (PDE) that govern most real-world problems, has become an indispensable tool for an engineer in various design stages. A more recent trend herein is to use the ever increasing computing power incorporate uncertainty and variability, which is omnipresent is all real-live applications, into these FE models. Several advanced techniques for incorporating either variability between nominally identical parts or spatial variability within one part into the FE models, have been introduced in this context. For the representation of spatial variability on the parameters of an FE model in a possibilistic context, the theory of Interval Fields (IF) was proven to show promising results. Following this approach, variability in the input FE model is introduced as the superposition of base vectors, depicting the spatial ‘patterns’, which are scaled by interval factors, which represent the actual variability. Application of this concept, however, requires identification of the governing parameters of these interval fields, i. e. the base vectors and interval scalars. Recent work of the authors therefore was focussed on finding a solution to the inverse problem, where the spatial uncertainty on the output side of the model is known from measurement data, but the spatial variability on the input parameters is unknown. Based on an a priori knowledge on the constituting base vectors of the interval field, the simulated output of the IFFEM computation is compared to measured data, and the input parameters are iteratively adjusted in order to minimize the discrepancy between the variability in simulation and measurement data. This discrepancy is defined based on geometric properties of the convex sets of both measurement and simulation data. However, the robustness of this methodology with respect to the size of the measurement data set that is used for the identification, as yet remains unclear. This paper therefore is focussed on the investigation of this robustness, by performing the identification on different measurement sets, depicting the same variability in the dynamic response of a simple FE model, which contain a decreasing amount of measurement replica. It was found that accurate identification remains feasible, even under a limited amount of measurement replica, which is highly relevant in the context of a non-probabilistic representation of variability in the FE model parameters. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic multiobjective acceptability analysis for the Choquet integral preference model and the scale construction problem

The Choquet integral preference model is adopted in Multiple Criteria Decision Aiding (MCDA) to deal with interactions between criteria, while the Stochastic Multiobjective Acceptability Analysis (SMAA) is an MCDA methodology considered to take into account uncertainty or imprecision on the considered data and preference parameters. In this paper, we propose to combine the Choquet integral preference model with the SMAA methodology in order to get robust recommendations taking into account all parameters compatible with the preference information provided by the Decision Maker (DM). In case the criteria are on a common scale, one has to elicit only a set of non-additive weights, technically a capacity, compatible with the DM’s preference information. Instead, if the criteria are on different scales, besides the capacity, one has to elicit also a common scale compatible with the preferences given by the DM. Our approach permits to explore the whole space of capacities and common scales compatible with the DM’s preference information.     

An outranking method under uncertainty

The uncertainty of consequences and the imprecision of data often imply, in multicriteria decision problems, the use of probability distributions to characterize the evaluation of each action with respect to eacg criterion. To keep as much information as possible, the analysis should treat directly these probability distributions instead of reducing them to single values such as mean or median. In this context, the paper proposes a multicriteria procedure which transforms these distributive evaluations of actions, according to decisionmaker's preferences, in order to progress to a ranking of these actions. The procedure consists, for each couple of actions, to construct a distributive preference degree with respect to each criterion and a distributive outranking degree over all criteria. These distributive outranking degrees are then explored in order to rank the actions, totally or partially.     

Modeling uncertain activity duration by fuzzy number and discrete-event simulation

This paper introduces how to incorporate fuzzy set theory and a fuzzy ranking measure with discrete-event simulation in order to model uncertain activity duration in simulating a real-world system, especially when insufficient or no sample data are available. Fuzzy numbers are used to describe uncertain activity durations, reflecting vagueness, imprecision and subjectivity in the estimation of them. A fuzzy ranking measure is merged with an activity scanning simulation algorithm for performing fuzzy simulation time advancement and event selection for simulation experimentation. The uses of the fuzzy activity duration and the probability distribution-modeled duration are compared through a series of simulation experiments. It is observed that the fuzzy simulation outputs are arrived at through only one cycle of fuzzy discrete-event simulation, still they contain all the statistical information that are produced through multiple cycles of simulation experiments when the probability distribution approach is adopted.     

Three Value Estimates of Input Variables and Health Manpower Projections

The introduction of variability and stochastic processes in health manpower projections may help health planners cope with the inherent problem of uncertainty in the future. However, limited data and the complex nature of health manpower make it very difficult or even impossible to estimate the probability distribution of input variables. In this paper, two statistical methods are discussed and compared for approximating a probability distribution based on imperfect data. The common feature of the two methods is that they use minimum, maximum, and most likely values, which can be estimated by people with little knowledge of statistics and probability. In addition, the methods can be used to analyse variables with a symmetrical as well as non-symmetrical probability distribution. An example is provided of the application of the methods to health manpower projections in China.     

A subjective Bayesian approach to the theory of queues I — Modeling

This is the first of an expository two-part paper which outlines a point of view different from that currently used in queueing theory. In both parts, the focus is on concepts. Here we adopt a personal probability point of view to all sources of uncertainty in the theory of queues and explore the consequences of our approach by comparing our results to those that are currently available. For ease of exposition, we confine attention to the M/M/1/ and the M/M/1/K queues. In Part I we outline the general strategy and focus on model development. In Part II we address the problem of inference in queues within the subjective Bayesian paradigm and introduce a use of Shannon's measure of information for assessing the amount of information conveyed by the various types of data from queues.     

Analyzing the Sensitivity of Generalized Linear Models to Incomplete Outcomes via the IDE Algorithm

Incomplete data models typically involve strong untestable assumptions about the missing data distribution. As inference may critically depend on them, the importance of sensitivity analysis is well recognized. Molenberghs, Kenward, and Goetghebeur proposed a formal frequentist approach to sensitivity analysis which distinguishes ignorance due to unintended incompleteness from imprecision due to finite sampling by design. They combine both sources of variation into uncertainty. This article develops estimation tools for ignorance and uncertainty concerning regression coefficients in a complete data model when some of the intended outcome values are missing. Exhaustive enumeration of all possible imputations for the missing data requires enormous computational resources. In contrast, when the boundary of the occupied region is of greatest interest, reasonable computational efficiency may be achieved via the imputation towards directional extremes (IDE) algorithm. This is a special imputation method designed to mark the boundary of the region by maximizing the direction of change of the complete data estimator caused by perturbations to the imputed outcomes. For multi-dimensional parameters, a dimension reduction approach is considered. Additional insights are obtained by considering structures within the region, and by introducing external knowledge to narrow the boundary to useful proportions. Special properties hold for the generalized linear model. Examples from a Kenyan HIV study will illustrate the points.     

A modified insurance risk process with uncertainty

An insurance risk process is traditionally considered by describing the claim process via a renewal reward process and assuming the total premium to be proportional to the time with a constant ratio. It is usually modeled as a stochastic process such as the compound Poisson process, and historical data are collected and employed to estimate the corresponding parameters of probability distributions. However, there exists the case of lack of data such as for a new insurance product. An alternative way is to estimate the parameters based on experts’ subjective belief and information. Therefore, it is necessary to employ the uncertain process to model the insurance risk process. In this paper, we propose a modified insurance risk process in which both the claim process and the premium process are assumed to be renewal reward processes with uncertain factors. Then we give the inverse uncertainty distribution of the modified process at each time. On this basis, we derive the ruin index which has an explicit expression based on given uncertainty distributions.     

Cure Cycle Design for Thermosetting-Matrix Composites Fabrication under Uncertainty

Design of the optimal cure temperature cycle is imperative for low-cost of manufacturing thermosetting-matrix composites. Uncertainties exist in several material and process parameters, which lead to variability in the process performance and product quality. This paper addresses the problem of determining the optimal cure temperature cycles under uncertainty. A stochastic model is developed, in which the parameter uncertainties are represented as probability density functions, and deterministic numerical process simulations based on the governing process physics are used to determine the distributions quantifying the output parameter variability. A combined Nelder–Mead Simplex method and the simulated annealing algorithm is used in conjunction with the stochastic model to obtain time-optimal cure cycles, subject to constraints on parameters influencing the product quality. Results are presented to illustrate the effects of a degree of parameter uncertainty, constraint values, and material kinetics on the optimal cycles. The studies are used to identify a critical degree of uncertainty in practice above which a rigorous analysis and design under uncertainty is warranted; below this critical value, a deterministic optimal cure cycle may be used with reasonable confidence.     

Reasoning with imprecise belief structures

This paper extends the theory of belief functions by introducing new concepts and techniques, allowing to model the situation in which the beliefs held by a rational agent may only be expressed (or are only known) with some imprecision. Central to our approach is the concept of interval-valued belief structure (IBS), defined as a set of belief structures verifying certain constraints. Starting from this definition, many other concepts of Evidence Theory (including belief and plausibility functions, pignistic probabilities, combination rules and uncertainty measures) are generalized to cope with imprecision in the belief numbers attached to each hypothesis. An application of this new framework to the classification of patterns with partially known feature values is demonstrated.     

The extended linear assignment method for multiple criteria decision analysis based on interval-valued intuitionistic fuzzy sets

The theory of interval-valued intuitionistic fuzzy sets is useful and beneficial for handling uncertainty and imprecision in multiple criteria decision analysis. In addition, the theory allows for convenient quantification of the equivocal nature of human subjective assessments. In this paper, by extending the traditional linear assignment method, we propose a useful method for solving multiple criteria evaluation problems in the interval-valued intuitionistic fuzzy context. A ranking procedure consisting of score functions, accuracy functions, membership uncertainty indices, and hesitation uncertainty indices is presented to determine a criterion-wise preference of the alternatives. An extended linear assignment model is then constructed using a modified weighted-rank frequency matrix to determine the priority order of various alternatives. The feasibility and applicability of the proposed method are illustrated with a multiple criteria decision-making problem involving the selection of a bridge construction method. Additionally, a comparative analysis with other methods, including the approach with weighted aggregation operators, the closeness coefficient-based method, and the auxiliary nonlinear programming models, has been conducted for solving the investment company selection problem to validate the effectiveness of the extended linear assignment method.