Where do we stand on measures of uncertainty,ambiguity, fuzziness,and the like?

It is argued in this paper that the theory of fuzzy sets involves at least four fundamentally different types of uncertainty. Each of these types requires a measure by which the degree of uncertainty of that type can be determined.Two main categories of uncertainty are connected with the terms ‘vagueness’ (or ‘fuzziness’) and ‘ambiguity’. In general, vagueness is associated with the difficulty of making sharp or precise distinctions in the world. Ambiguity, on the other hand, is associated with one-to-many relations, i.e., situations with two or more alternatives that are left unspecified. While the concept of a fuzzy set represents a basic mathematical framework for dealing with vagueness, the concept of a fuzzy measure is a general framework for dealing with ambiguity.Several classes of measures of vagueness, usually referred to as measures of fuzziness, have been proposed in the literature. Each class is based on some underlying conception of the degree of fuzziness. A general set of requirements for measures of fuzziness is formulated, followed by an overview of the measures proposed in the literature.Measures of ambiguity are discussed within the framework of plausibility and belief measures. Although it does not cover all fuzzy measures, this framework is sufficiently broad for most practical purposes, and represents a generalization of both probability theory and possibility theory.It is argued that three complementary measures of ambiguity should be employed. One of them is obtained by generalizing the Hartley measure of uncertainty; it measures the degree of nonspecificity in individual situations described by the various belief and plausibility measures. The other two are obtained by generalizing the well known Shannon measure of uncertainty; they measure the degree of dissonance and the degree of confusion in evidence, respectively. Basic mathematical properties of these measures are overviewed.It is also argued that each of the four types of uncertainty measures, which are fundamentally different from each other, can be used for measuring structural (syntactic) information in the same sense as the Hartley and Shannon measures have been used in this respect. As such, these measures are potentially powerful tools for dealing with systems problems such as systems modelling, analysis, simplification, or design.     

An Inverse Reliability-based Approach for Designing under Uncertainty with Application to Robust Piston Design

In this work, we propose an optimization framework for designing under uncertainty that considers both robustness and reliability issues. This approach is generic enough to be applicable to engineering design problems involving nonconvex objective and constraint functions defined in terms of random variables that follow any distribution. The problem formulation employs an Inverse Reliability Strategy that uses percentile performance to address both robustness objectives and reliability constraints. Robustness is achieved through a design objective that evaluates performance variation as a percentile difference between the right and left trails of the specified goals. Reliability requirements are formulated as Inverse Reliability constraints that are based on equivalent percentile performance levels. The general proposed approach first approximates the formulated problem via a Gaussian Kriging model. This is then used to evaluate the percentile performance characteristics of the different measures inherent in the problem formulation for various design variable settings via a Most Probable Point of Inverse Reliability search algorithm. By using these percentile evaluations in concert with the response surface methodology, a polynomial programming approximation is generated. The resulting problem formulation is finally solved to global optimality using the Reformulation–Linearization Technique (RLT) approach. We demonstrate this overall proposed approach by applying it to solve the problem of reducing piston slap, an undesirable engine noise due to the secondary motion of a piston within a cylinder.     

Extending the use of scenario planning and MCDA for the evaluation of strategic options

Multi-criteria decision analysis (MCDA) is well equipped to deal with conflicting, qualitative objectives when evaluating strategic options. Scenario planning provides a framework for confronting uncertainty, which MCDA lacks. Integration of these methods offers various advantages, yet its effective application in evaluating strategic options would benefit from scenarios that reflect a larger number of wide-ranging scenarios developed in a time-efficient manner, as well as incorporation of MCDA measures that inform within and across scenario comparison of options. The main contribution of this paper is to illustrate how a more diverse set of scenarios could be developed quickly, and to investigate how regret could be used to facilitate comparison of options. First, the reasons for these two areas of development are elaborated with respect to existing techniques. The impacts of applying the proposed method in practice are then assessed through a case study involving food security in Trinidad and Tobago. The paper concludes with a discussion of findings and areas for further research.     

Extension of quantum information by using entropy of deterministic functions

The concept of entropy of random variables first defined by Shannon has been generalized later in various ways by mathematicians who so obtained new measures of uncertainty, again for random variables. Recently, the author suggested another extension which provides a meaningful definition for the entropy of deterministic functions, both in the sense of Shannon and of Renyi. These measures of uncertainty are different from those which are utilized by physicists in the study of chaotic dynamics, like the Kolmogorov entropy for instance.

The aim of this paper is to go a step further, and to derive measures of uncertainty for operators, by using exactly the same rationale. After a short background on the entropies of deterministic functions, one obtains successively the entropy of a constant square matrix operator, the entropy of a varying square matrix operator, the entropy of the kernel of an integral transformation, and the entropy of differential operators defined by square matrices.

Then one carefully exhibits the relation which exists between these results and the quantum mechanical entropy first introduced by Von Neumann, and one so obtains a new generalized quantum mechanical entropy which applies to matrics which are not necessarily density matrices. Finally, some illustrative examples for future applications are outlined.     

Toward a general theory of fuzzy variables

A general framework for a theory is presented that encompasses both statistical uncertainty, which falls within the province of probability theory, and nonstatistical uncertainty, which relates to the concept of a fuzzy set and possibility theory [L. A. Zadeh, J. Fuzzy Sets1 (1978), 3–28]. The concept of a fuzzy integral is used to define the expected value of a random variable. Properties of the fuzzy expectation are stated and a mean-value theorem for the fuzzy integral is proved. Comparisons between the fuzzy and the Lebesgue integral are presented. After a new concept of dependence is formulated, various convergence concepts are defined and their relationships are studied by using a Chebyshev-like inequality for the fuzzy integral. The possibility of using this theory in Bayesian estimation with fuzzy prior information is explored.     

Safety Stock and Service Levels in Periodic Review Inventory Systems

The management of requirement (purchased parts and components) uncertainty in MRP systems is a serious problem for manufacturing firms. When manufacturing service levels fall below a predetermined service level, safety stock must be increased in order to adjust to the competitive environment. This paper presents the results of a set of simulation experiments in which we investigate some of the complexities of safety stock requirements, lot sizing performance and the marginal safety stock requirements associated with various service level policies. The results of the experiments have empirically shown that the marginal units of safety stock required for various service level policies follow a predictable (measurable) pattern. The research also indicates that the economic advantages of well-known dynamic lot sizing procedures are diminished by the presence of requirements uncertainty. Another important finding is the interaction between service level and ‘lumpy’ requirements. High levels of the coefficient of variation and the economic time between orders resulted in larger orders, reduced safety stock and high service levels. In general, the overall findings from this research will hopefully provide the decision maker with an in-depth understanding of the measurement and impact of various service level policies.     

Models for planning capacity expansion of convenience stores under uncertain demand and the value of information

Several stochastic optimization models for planning capacity expansion for convenience store chains (or other similar businesses) are developed that incorporate uncertainty in future demand. All of these models generate schedules for capacity expansion, specifying the size, location, and timing of these expansions in order to maximize the expected profit to the company and to remain within a budget constraint on available resources. The models differ in how uncertainty is incorporated, specifically they differ in the point in the decision-making process that the uncertainty in the demand is resolved. Several measures of the value of information are defined by comparing the results from the different models. Two sample problems are given and their solutions for the various approaches compared.     

A rolling horizon approach for disruption management of railway rolling stock

This paper deals with real-time disruption management of rolling stock in passenger railway transportation. We describe a generic framework for dealing with disruptions of railway rolling stock schedules. The framework is presented as an online combinatorial decision problem, where the uncertainty of a disruption is modeled by a sequence of information updates. To decompose the problem and to reduce the computation time, we propose a rolling horizon approach: rolling stock decisions are only considered if they are within a certain time horizon from the time of rescheduling. The schedules are then revised as time progresses and new information becomes available. We extend an existing model for rolling stock scheduling to the specific requirements of the real-time situation, and we apply it in the rolling horizon framework. We perform computational tests on instances constructed from real-life cases of Netherlands Railways (NS), the main operator of passenger trains in the Netherlands. We explore the consequences of different settings of the approach for the trade-off between solution quality and computation time.     

The daily tail assignment problem under operational uncertainty using look-ahead maintenance constraints

The tail assignment problem is a critical part of the airline planning process that assigns specific aircraft to sequences of flights, called lines-of-flight, to satisfy operational constraints. The aim of this paper is to develop an operationally flexible method, based upon the one-day routes business model, to compute tail assignments that satisfy short-range—within the next three days—aircraft maintenance requirements. While maintenance plans commonly span multiple days, the methods used to compute tail assignments for the given plans can be overly complex and provide little recourse in the event of schedule perturbations. The presented approach addresses operational uncertainty by using solutions from the one-day routes aircraft maintenance routing approach as input. The daily tail assignment problem is solved with an objective to satisfy maintenance requirements explicitly for the current day and implicitly for the subsequent two days. A computational study will be performed to assess the performance of exact and heuristic solution algorithms that modify the input lines-of-flight to reduce maintenance misalignments. The daily tail assignment problem and the developed algorithms are demonstrated to compute solutions that effectively satisfy maintenance requirements when evaluated using input data collected from three different airlines.     

On weighted interval entropy

Measures of uncertainty in past and residual lifetime distributions have been proposed in the information-theoretic literature. Recently, Di Crescenzo and Longobardi (2006) introduced weighted differential entropy and its dynamic versions. These information-theoretic uncertainty measures are shift-dependent. In this paper, we study the weighted differential information measure for two-sided truncated random variables. This new measure is a generalization of recent dynamic weighted entropy measures. We study various properties of this measure, including its connection with weighted residual and past entropies, and we obtain its upper and lower bounds.     

A probabilistic model applied to emergency service vehicle location

This paper is concerned with the formulation and the solution of a probabilistic model for determining the optimal location of facilities in congested emergency systems. The inherent uncertainty which characterizes the decision process is handled by a new stochastic programming paradigm which embeds the probabilistic constraints within the traditional two-stage framework. The resulting model drops simplifying assumptions on servers independence allowing at the same time to handle the spatial dependence of demand calls. An exact solution method and different tailored heuristics are presented to efficiently solve the problem. Computational experience is reported with application to various networks.     

Superiority–inferiority modeling coupled minimax-regret analysis for energy management systems

In this study, a superiority–inferiority-based minimax-regret analysis (SI-MRA) model is developed for supporting the energy management systems (EMS) planning under uncertainty. In SI-MRA model, techniques of fuzzy mathematical programming (FMP) with the superiority and inferiority measures and minimax regret analysis (MMR) are incorporated within a general framework. The SI-MRA improves upon conventional FMP methods by directly reflecting the relationships among fuzzy coefficients in both the objective function and constraints with a high computational efficiency. It can not only address uncertainties expressed as fuzzy sets in both of the objective function and system constraints but also can adopt a list of scenarios to reflect the uncertainties of random variables without making assumptions on their possibilistic distributions. The developed SI-MRA model is applied to a case study of long-term EMS planning, where fuzziness and randomness exist in the costs for electricity generation and demand. A number of scenarios associated with various alternatives and outcomes under different electricity demand levels are examined. The results can help decision makers identify an optimal strategy of planning electricity generation and capacity expansion based on a minimax regret level under uncertainty.     

Incorporating demand uncertainty and forecast error in supply chain planning models

This paper develops a framework for examining the effect of demand uncertainty and forecast error on unit costs and customer service levels in the supply chain, including Material Requirements Planning (MRP) type manufacturing systems. The aim is to overcome the methodological limitations and confusion that has arisen in much earlier research. To illustrate the issues, the problem of estimating the value of improving forecasting accuracy for a manufacturer was simulated. The topic is of practical importance because manufacturers spend large sums of money in purchasing and staffing forecasting support systems to achieve more accurate forecasts. In order to estimate the value a two-level MRP system with lot sizing where the product is manufactured for stock was simulated. Final product demand was generated by two commonly occurring stochastic processes and with different variances. Different levels of forecasting error were then introduced to arrive at corresponding values for improving forecasting accuracy. The quantitative estimates of improved accuracy were found to depend on both the demand generating process and the forecasting method. Within this more complete framework, the substantive results confirm earlier research that the best lot sizing rules for the deterministic situation are the worst whenever there is uncertainty in demand. However, size matters, both in the demand uncertainty and forecasting errors. The quantitative differences depend on service level and also the form of demand uncertainty. Unit costs for a given service level increase exponentially as the uncertainty in the demand data increases. The paper also estimates the effects of mis-specification of different sizes of forecast error in addition to demand uncertainty. In those manufacturing problems with high demand uncertainty and high forecast error, improved forecast accuracy should lead to substantial percentage improvements in unit costs. Methodologically, the results demonstrate the need to simulate demand uncertainty and the forecasting process separately.     

The stochastic ordering of mean-preserving transformations and its applications

The stochastic variability measures the degree of uncertainty for random demand and/or price in various operations problems. Its ordering property under mean-preserving transformation allows us to study the impact of demand/price uncertainty on the optimal decisions and the associated objective values. Based on Chebyshev’s algebraic inequality, we provide a general framework for stochastic variability ordering under any mean-preserving transformation that can be parameterized by a single scalar, and apply it to a broad class of specific transformations, including the widely used mean-preserving affine transformation, truncation, and capping. The application to mean-preserving affine transformation rectifies an incorrect proof of an important result in the inventory literature, which has gone unnoticed for more than two decades. The application to mean-preserving truncation addresses inventory strategies in decentralized supply chains, and the application to mean-preserving capping sheds light on using option contracts for procurement risk management.     

Discretizations of Linear Elliptic Partial Differential Inclusions

Differential inclusions provide a suitable framework for modelling choice and uncertainty. In finite dimensions, the theory of ordinary differential inclusions and their numerical approximations is well-developed, whereas little is known for partial differential inclusions, which are the deterministic counterparts of stochastic partial differential equations.

The aim of this article is to analyze strategies for the numerical approximation of the solution set of a linear elliptic partial differential inclusion. The geometry of its solution set is studied, numerical methods are proposed, and error estimates are provided.     

Modeling attitudes towards uncertainty and risk through the use of choquet integral

The aim of this paper is to present in a unified framework a survey of some results related to Choquet Expected Utility (CEU) models, a promising class of models introduced separately by Quiggin [35], Yaari [48] and Schmeidler [40, 41] which allow to separate attitudes towards uncertainty (or risk) from attitudes towards wealth, while respecting the first order stochastic dominance axiom.     

Simulation-based framework to improve patient experience in an emergency department

The global economic crisis has a significant impact on healthcare resource provision worldwide. The management of limited healthcare resources is further challenged by the high level of uncertainty in demand, which can lead to unbalanced utilization of the available resources and a potential deterioration of patient satisfaction in terms of longer waiting times and perceived reduced quality of services. Therefore, healthcare managers require timely and accurate tools to optimize resource utility in a complex and ever-changing patient care process. An interactive simulation-based decision support framework is presented in this paper for healthcare process improvement. Complexity and different levels of variability within the process are incorporated into the process modeling phase, followed by developing a simulation model to examine the impact of potential alternatives. As a performance management tool, balanced scorecard (BSC) is incorporated within the framework to support continual and sustainable improvement by using strategic-linked performance measures and actions. These actions are evaluated by the simulation model developed, whilst the trade-off between objectives, though somewhat conflicting, is analysed by a preference model. The preference model is designed in an interactive and iterative process considering decision makers preferences regarding the selected key performance indicators (KPIs). A detailed implementation of the framework is demonstrated on an emergency department (ED) of an adult teaching hospital in north Dublin, Ireland. The results show that the unblocking of ED outflows by in-patient bed management is more effective than increasing only the ED physical capacity or the ED workforce.     

Generalized information theory for hints

This paper develops a new uncertainty measure for the theory of hints that complies with the established semantics of statistical information theory and further satisfies all classical requirements for such a measure imposed in the literature. The proposed functional decomposes into conversant uncertainty measures and therefore discloses a new interpretation of the latters as well. By abstracting to equivalence classes of hints we transport the new measure to mass functions in Dempster-Shafer theory and analyse its relationship with the aggregate uncertainty, which currently is the only known functional for the Dempster-Shafer theory of evidence that satisfies the same set of properties. Moreover, the perspective of hints reveals that the standard independence notion in Dempster-Shafer theory called non-interactivity corresponds to an amalgamation of probabilistic independence and qualitative independence between frames of discernment. All results in this paper are developed for arbitrary families of compatible frames generalizing the very specialized multi-variate systems that are usually studied in information theory.