Uncertain Quantity Research Articles

Considering pricing and uncertainty in designing a reverse logistics network

Companies try to maximize their profits by recovering returned products of highly uncertain quality and quantity. In this paper, a reverse logistics network for an Original Equipment Manufacturer (OEM) is presented. Returned products are selected for remanufacturing or scrapping, based on their quality and proportional prices are offered to customers. A Mixed Integer Non-linear Programming (MINLP) model is proposed to determine the location of collection centers, the optimum price of returned products and the sorting policy. The risk in the objective function is measured using the Conditional Value at Risk (CVaR) metric. CVaR measures the risk of an investment in a conservative way by considering the maximum lost. The results are analyzed for various values of the risk parameters ({\alpha}, and {\lambda}). These parameters indicate that considering risk affects prices, the classification of returned products, the location of collection centers and, consequently, the objective function. The model performs more conservatively when the weight of the CVaR part ({\lambda}) and the value of the confidence level {\alpha} are increased. The results show that better profits are obtained when we take CVaR into account.

Multilevel Monte Carlo estimation of expected information gains

The expected information gain is an important quality criterion of Bayesian experimental designs, which measures how much the information entropy about uncertain quantity of interest θ is reduced on average by collecting relevant data Y. However, estimating the expected information gain has been considered computationally challenging since it is defined as a nested expectation with an outer expectation with respect to Y and an inner expectation with respect to θ. In fact, the standard, nested Monte Carlo method requires a total computational cost of to achieve a root-mean-square accuracy of ε. In this paper we develop an efficient algorithm to estimate the expected information gain by applying a multilevel Monte Carlo (MLMC) method. To be precise, we introduce an antithetic MLMC estimator for the expected information gain and provide a sufficient condition on the data model under which the antithetic property of the MLMC estimator is well exploited such that optimal complexity of is achieved. Furthermore, we discuss how to incorporate importance sampling techniques within the MLMC estimator to avoid arithmetic underflow. Numerical experiments show the considerable computational cost savings compared to the nested Monte Carlo method for a simple test case and a more realistic pharmacokinetic model.

Assessment of surface and bulk-dominated methodologies to measure critical resolved shear stresses in hexagonal materials

Crystallographic slip in hexagonal metals involves a number of geometrically distinct slip families characterized by their slip direction and slip plane (basal, prismatic, and pyramidal). Owing to the low symmetry of hexagonal lattices, each of these slip families only have a few symmetrically equivalent slip systems (family members). Furthermore, different slip families become active at different resolved shear stress, i.e., they have different critical resolved shear stress values (CRSS). The plastic anisotropy of hexagonal materials makes the numerical prediction of their plastic behavior challenging and depends critically on the knowledge of CRSS values. The present contribution assesses the reliability of three proposed methods (with additional variations) to quantify CRSS values of the different hexagonal slip families. Those methods (a to c) rely on: (a) the statistics of observed surface slip traces in a (slightly) deformed polycrystal; (b) an iterative adjustment of CRSS values until a simulated single crystal indentation matches the corresponding experiment in terms of load–displacement response and residual surface topography of the indent; (c) in-situ high-energy X-ray diffraction to measure the evolution of resolved stress (from lattice strains) in grains for which single-family slip is probable based upon specific lattice reorientation conditions. Virtual experiments are performed on synthetic microstructures such that the (extracted) CRSS values resulting from simulating the different methodologies can be rigorously compared against the (target) CRSS values that are installed in the phenomenological constitutive material description used in the simulations. The resulting CRSS values of methods (a) exhibit a strong dependence on, and deterioration with, decreasing level of slip trace observability, which is an uncertain quantity in experimental measurements. For the inverse indentation method (b), the predicted CRSS values are within 8 % of their reference CRSS values for the two investigated cases. The high-energy X-ray diffraction method (c) most reliably determines CRSS values for basal and prism slip, but lacks a strict grain selection criterion to assess pyramidal slip.

Application of Recurrent Neural Networks in the numerical analysis of reinforced concrete structures considering polymorphic uncertainty

AbstractThe realistic modelling of structures is fundamental for their numerical simulation and is mainly characterized by the physical approach and the consideration of the available data by an adequate uncertainty description. The key idea in the contribution at hand is the consideration of polymorphic uncertainty in the numerical structural analysis and in the mechanical modelling for reinforced concrete structures, which are characterized by a combination of heterogeneous concrete and different types of reinforcement (e.g. steel bars or woven carbon fibres). Typically, the reinforcement is determined by another length scale compared to the overall structural size. The formulation and development of a computational homogenization approach, considering the different homogeneous and heterogeneous characteristics of a macroscopic structure, are essential for a realistic numerical computation. In recent years, focal point of research was on structural analysis considering uncertain material or geometry parameters. Probabilistic approaches are dominating the uncertainty consideration currently, although they are associated with specific disadvantages and limitations. In this contribution, a generalized uncertainty model is utilized in order to take variability, impression as well as inaccuracy, vagueness and incompleteness into account. This allows a separated evaluation of the influence for each uncertainty source on the results. Therefore, polymorphic uncertainty models are applied and developed by combining aleatoric and epistemic uncertainty descriptions, resulting e.g. in the formulation of the uncertainty model “fuzzy probability based randomness”. The information of the different length scales is considered to be uncertain, e.g. the geometry or the material properties of a representative volume element (RVE) at the meso scale. Subsequently, the uncertainty of a macro structure is derived from uncertain results on the meso structure. In the contribution, a parametrized RVE for concrete structures, including cement phase as well as aggregates, is presented. Various material parameters are considered as uncertain, which results in uncertain effective quantities. Assuming, that an uncertain quantity is substitutable by a combination of a representative measure and multiple uncertainty characterizing measures, an approximation of both type of quantities by a recurrent neural network is carried out. The surrogate models are utilized as constitutive description in a numerical structural analysis on macro level, providing uncertain structural responses.

Circular economy implementation in waste management network design problem: a case study

The paper presents a new approach to support the strategic decision-making in the area of municipal solid waste management applying modern circular economy principles. A robust two-stage integer non-linear program is developed. The primary goal tends to reduce the waste production. The generated waste should be preferably recycled as much as possible and the resultant residual waste might be used for energy recovery. Only some waste residues are appropriate for landfilling. The aim is to propose the near-optimal waste allocation for its suitable processing as well as waste transportation plan at an operational level. In addition, the key strategical decisions on waste treatment facilities location must be made. Since waste production is very often hard to predict, it is modeled as an uncertain decision-dependent quantity. To support the circular economy ideas, advertising and pricing principles are introduced and applied. Due to the size of available real-world data and complexity of the designed program, the presented model is linearized and uncertainty is handled by a robust optimization methodology. The model, data, and algorithm are implemented in MATLAB and Julia, using the state-of-the-art solvers. The computational result is a set of decisions providing a trade-off between the average performance and the immunization against the worst-case conditions.

Robust design optimization for egressing pedestrians in unknown environments

In this paper, we deal with a size-variable group of pedestrians moving in a unknown confined environment and searching for an exit. Pedestrian dynamics are simulated by means of a recently introduced microscopic (agent-based) model, characterized by an exploration phase and an egress phase. First, we study the model to reveal the role of its main parameters and its qualitative properties. Second, we tackle a robust optimization problem by means of the Particle Swarm Optimization method, aiming at reducing the time-to-target by adding in the walking area multiple obstacles optimally placed and shaped. Robustness is sought against the number of people in the group, which is an uncertain quantity described by a random variable with given probability density distribution.

Expert Knowledge Elicitation: Subjective but Scientific

ABSTRACT Expert opinion and judgment enter into the practice of statistical inference and decision-making in numerous ways. Indeed, there is essentially no aspect of scientific investigation in which judgment is not required. Judgment is necessarily subjective, but should be made as carefully, as objectively, and as scientifically as possible. Elicitation of expert knowledge concerning an uncertain quantity expresses that knowledge in the form of a (subjective) probability distribution for the quantity. Such distributions play an important role in statistical inference (for example as prior distributions in a Bayesian analysis) and in evidence-based decision-making (for example as expressions of uncertainty regarding inputs to a decision model). This article sets out a number of practices through which elicitation can be made as rigorous and scientific as possible. One such practice is to follow a recognized protocol that is designed to address and minimize the cognitive biases that experts are prone to when making probabilistic judgments. We review the leading protocols in the field, and contrast their different approaches to dealing with these biases through the medium of a detailed case study employing the SHELF protocol. The article ends with discussion of how to elicit a joint probability distribution for multiple uncertain quantities, which is a challenge for all the leading protocols. Supplementary materials for this article are available online.

ASEAN grid flexibility: Preparedness for grid integration of renewable energy

In 2015, ASEAN established a goal of increasing its renewable energy share in its energy portfolio from approximately 13–23% by 2025. Renewable electricity, especially intermittent and variable sources, presents challenges for grid operators due to the uncertain timing and quantity of electricity supply. Grid flexibility, the electric grid's ability to respond to changing demands and supply, now stands a key resource in responding to these uncertainties while maximizing the cost-effective role of clean energy. We develop and apply a grid flexibility assessment tool to assess ASEAN's current grid flexibility using six quantitative indicators: grid reliability, electricity market access; load profile ramp capacity; quality of forecasting tools; proportion of electricity generation from natural gas; and renewable energy diversity. We find that ASEAN nations cluster into three groups: better; moderately; and the least prepared nations. We develop an analytical ramp rate calculator to quantify expected load ramps for ASEAN in an integrated ASEAN Power Grid scenario. The lack of forecasting systems and limited electricity market access represent key weaknesses and areas where dramatic improvements can become cost-effective means to increase regional grid flexibility. As ASEAN pursues renewable energy targets, regional cooperation remains essential to address identified challenges. Member nations need to increase grid flexibility capacity to adequately prepare for higher penetrations of renewable electricity and lower overall system costs.

Assessment of Surface and Bulk-Dominated Methodologies to Measure Critical Resolved Shear Stresses in Hexagonal Materials

Crystallographic slip in hexagonal metals involves a number of geometrically distinct slip families characterized by their slip direction and slip plane (basal, prismatic, and pyramidal). Owing to the low symmetry of hexagonal lattices, each of these slip families only have a few symmetrically equivalent slip systems (family members). Furthermore, different slip families become active at different resolved shear stress, i. e., they have different critical resolved shear stress values (CRSS). The plastic anisotropy of hexagonal materials renders the numerical prediction of their plastic behavior challenging and depends critically on the knowledge of CRSS values. The present contribution assesses the reliability of three proposed methods (with additional variations) to quantify CRSS values of the different hexagonal slip families. Those methods (a to c) rely on: (a) the statistics of observed surface slip traces in a (slightly) deformed polycrystal; (b) an iterative adjustment of CRSS values until a simulated single crystal indentation matches the corresponding experiment in terms of load-displacement response and residual surface topography of the indent; (c) in-situ high-energy X-ray diffraction to measure the evolution of resolved stress (from lattice strains) in grains for which single family slip can be deduced from specific lattice reorientation conditions. Virtual experiments are performed on synthetic microstructures to assess how reliably each method recovers prescribed CRSS values using a phenomenological constitutive material description. The resulting CRSS values of methods (a) exhibit a strong dependence on, and deteriorates with decreasing level of slip trace observability, which is an uncertain quantity in experimental measurements. For the inverse indentation method (b), the predicted CRSS values are within 8% of their reference CRSS values for the two investigated cases. The high-energy X-ray diffraction method (c) most reliably determines CRSS values for basal and prism slip, but lacks a strict grain selection criterion to assess pyramidal slip.

Assessment of the response of pollinator abundance to environmental pressures using structured expert elicitation

Policy-makers often need to rely on experts with disparate fields of expertise when making policy choices in complex, multi-faceted, dynamic environments such as those dealing with ecosystem services. For policy-makers wishing to make evidence-based decisions which will best support pollinator abundance and pollination services, one of the problems faced is how to access the information and evidence they need, and how to combine it to formulate and evaluate candidate policies. This is even more complex when multiple factors provide influence in combination. The pressures affecting the survival and pollination capabilities of honey bees (Apis mellifera), wild bees, and other pollinators are well documented, but incomplete. In order to estimate the potential effectiveness of various candidate policy choices, there is an urgent need to quantify the effect of various combinations of factors on the pollination ecosystem service. Using high-quality experimental evidence is the most robust approach, but key aspects of the system may not be amenable to experimentation or may be prohibitive based on cost, time and effort. In such cases, it is possible to obtain the required evidence by using structured expert elicitation, a method for quantitatively characterizing the state of knowledge about an uncertain quantity. Here we report and discuss the outputs of the novel use of a structured expert elicitation, designed to quantify the probability of good pollinator abundance given a variety of weather, disease, and habitat scenarios.Evaluación de la respuesta de la abundancia de polinizadores a las presiones ambientales mediante el uso de elicitación experta estructuradaA menudo los legisladores dependen de expertos en diversas áreas de conocimiento para tomar decisiones sobre legislación en entornos complejos, multifacéticos y dinámicos tales como los que tienen que ver con los servicios ecosistémicos. Los legisladores que quieren tomar decisiones basadas en evidencias que respalden mejor los servicios de polinización y la abundancia de polinizadores, se enfrentan al problema de cómo acceder a la información y a las evidencias que necesitan, y de cómo combinar éstas para formular y evaluar futuras leyes. Esto es aún más complejo cuando hay múltiples factores que influyen de manera combinada. Las presiones que afectan a la supervivencia y a la capacidad polinizadora de las abejas de la miel (Apis mellifera), a las abejas silvestres y a otros polinizadores están bien documentadas, pero de manera incompleta. Para estimar la efectividad potencial de varias opciones posibles de legislación, es necesario cuantificar el efecto combinado de varios factores sobre el servicio ecosistémico de polinización. El uso de una evidencia experimental de alta calidad es el enfoque más sólido, pero algunos aspectos clave del sistema podrían no ser susceptibles de experimentación o ser prohibitivos debido al coste, el tiempo y el esfuerzo. En tales casos, es posible obtener la evidencia requerida mediante el uso de la elicitación experta estructurada, un método para caracterizar cuantitativamente el estado del conocimiento sobre una cantidad incierta. En este estudio informamos y discutimos los resultados del uso novedoso de una elicitación experta estructurada, diseñada para cuantificar la probabilidad de una abundancia de polinizadores adecuada teniendo en cuenta una variedad de escenarios climáticos, de enfermedades y de hábitat.

EES-EOQ model with uncertain acquisition quantity and market demand in dedicated or combined remanufacturing systems

• Build up a mathematical model analyzing environmental benefits of remanufacture over manufacture. • Overcome the limitations of the classical EOQ model and assume acquisition and demand are both uncertain. • Prove the existence and uniqueness of optimal strategy mathematically. • Close-form expression of optimal solutions that maximize the benefits and minimize costs. • Optimal strategies for dedicated and combined remanufacturing models. The classical economic order quantity (EOQ) model assumes that market demand is deterministic and would be invalid without this premise. Thus far, available literature on the EOQ model do not consider any situation where both market demand and acquisition quantity are uncertain. Therefore, this work develops an evolved environmental sustainability EOQ (EES-EOQ) model to overcome the limitations of the classical EOQ model. Employing the remanufacture-up-to ratio to control the production quantity indirectly, we deduce three optimal strategies: (I) minimize the inventory cost, (II) maximize the overall environmental benefit in a dedicated remanufacturing model, and (III) coordinate forward logistics and reverse logistics to achieve environment optimization in the combined remanufacturing model. With stochastic analysis, we mathematically prove the existence and uniqueness of the optimal remanufacture-up-to ratio in each strategy. Validated by numerical experiments and sensitivity analysis, we demonstrate that the replenishment of new products in the combined model can effectively eliminate the risks of being out of stock and strike an optimal balance between uncertain supply and demand. This study provides the theoretical foundation for remanufacturers to develop optimal remanufacturing strategies for mitigating environmental issues caused by excessive carbon emissions.

Energy Recovery from the Organic Fraction of Municipal Solid Waste: A Real Options-Based Facility Assessment

During the last years, due to the strict regulations on waste landfilling, anaerobic digestion (AD) of the organic fraction of municipal solid waste (OFMSW) is increasingly considered a sustainable alternative for waste stabilization and energy recovery. AD can reduce the volume of OFMSW going to landfill and produce, at the same time, biogas and compost, all at a profit. The uncertainty about the collected quantity of organic fraction, however, may undermine the economic-financial sustainability of such plants. While the flexibility characterizing some AD technologies may prove very valuable in uncertain contexts since it allows adapting plant capacity to changing environments, the investment required for building flexible systems is generally higher than the investment for dedicated equipment. Hence, an adequate justification of investments in these flexible systems is needed. This paper presents the results of a study aimed at investigating how different technologies may perform from technical, economic and financial standpoints, in presence of an uncertain organic fraction quantity to be treated. Focusing on two AD treatment plant configurations characterized by a technological process with different degree of flexibility, a real options-based model is developed and then applied to the case of the urban waste management system of the Metropolitan Area of Bari (Italy). Results show the importance of pricing the flexibility of treatment plants, which becomes a critical factor in presence of an uncertain organic fraction. Hence, it has to be taken into consideration in the design phase of these plants.

Quantile Evaluation, Sensitivity to Bracketing, and Sharing Business Payoffs

From forecasting competitions to conditional value-at-risk requirements, the use of multiple quantile assessments is growing in practice. To evaluate them, we use a rule from the general class of proper scoring rules for a forecaster’s multiple quantiles of a single uncertain quantity of interest. The general rule is additive in the component scores. Each component contains a function that measures its quantile’s distance from the realization and weights its contribution to the overall score. To determine this function, we propose that the score of a group’s combined quantile should be better than that of a randomly selected forecaster’s quantile only when the forecasters bracket the realization (i.e., their quantiles do not fall on the same side of the realization). If a score satisfies this property, we say it is sensitive to bracketing. We characterize the class of proper scoring rules that is sensitive to bracketing when the decision maker uses a generalized average to combine forecasters’ quantiles. Finally, we show how weights can be set to match the payoffs in many important business contexts.

An inventory model with uncertain demand and lost sales reduction under service level constraint

In practice, the quantity received may not match with the quantity ordered, due to various reasons such as rejection during inspection, damage or breakage during transportation, human errors in counting etc. Subsequently, the managers often must make decisions under uncertain quantity received circumstances. This paper explores the feasibility of reducing ordering cost and lost sales caused by stock-out. In this study, we examine the continuous review inventory model with shortages include the case where the quantity received is uncertain in which the lead time and lost sales rate are decision variables and also with service level constraint. Here we consider the lead time crashing cost is a function of negative exponential lead time. The objective of this study is to minimize the joint expected total cost by simultaneously optimizing the order quantity and lost sales rate. An efficient algorithm for finding the optimal solution is developed and numerical examples are given to illustrate the results.

Supply risk sharing in a closed-loop supply chain

In closed-loop supply chains, the acquisition process of used products or materials often faces high volatility and results in both the uncertain quantity and quality of the recycled products. Such supply uncertainties happen in the reverse supply chain flows while the forward supply chain still faces market demand uncertainty. Managing these uncertainties while coordinating the reverse and forward supply chain flows is one of the many challenges faced by CLSC managers. Among different supply chain management approaches, supply chain contract design has been proven as an effective method for improving supply chain performance. Based on the supply chain contract design theories and focusing on the acquisition process in closed-loop supply chains, this paper analyzes different supply risk sharing contracts including under-supply risk sharing contract, over-supply risk sharing contract, complete supply risk sharing contract, and no risk sharing contract. With game-theoretical models, equilibrium between the recycling price decision (controlling the reverse flow) and the remanufacturing quantity decision (controlling the forward flow) are derived and analyzed. Further, numerical examples are used to demonstrate the managerial results. Based on both the theoretic results and the numerical example illustration, it is shown that different supply risk sharing contracts may result in both the recycling price decision and the remanufacturing production quantity decision to deviate from those decisions under centralized collection structure. Further, this paper investigates the impact of supply or demand risk reduction on the CLSC's financial and environmental performances. It is observed that supply risk reduction approaches may result in the alignment between the CLSC's financial and environmental goals while the demand risk reduction approaches cannot achieve such alignment.

Reliability updating in linear opinion pooling for multiple decision makers

Accurate information sources are vital prerequisites for good decision making. Here, we consider a multiple participant setting, where all decision makers have a collection of neighbours with whom they share their beliefs about some common relevant uncertain quantity. When determining which course of action to follow, a decision maker takes into account all the information received from her neighbours. Over time, in light of the returns observed from choices made, decision makers update their own beliefs over the uncertain event and also adjust the degree of consideration they afford to the opinions of each neighbour based on the level of reliability their information is ascertained to have. In this article, we derive and discuss a methodology, the plug-in approach, by which this can be done, and discuss some suitable justification for this. We include an illustrative risk-based example which further demonstrates the merits of the method, before concluding with a discussion on further work and potential alternatives.

Hierarchy evaluation of water resources vulnerability under climate change in Beijing, China

Rapid population growth and increased economic activity impose an urgent challenge on the sustainability of water resources in Beijing. Water resources system is a complex uncertain system under climate change which is of vulnerability. But water resources system vulnerability research is relatively weak. In this study, we present a multifunctional hierarchy indicator system for the performance evaluation of water resources vulnerability (WRV) under climate change. We established an evaluation model, i.e., analytic hierarchy process combining set pair analysis (AHPSPA) model, for assessing WRV, in which weight is determined by the analytic hierarchy process (AHP) method and the evaluation degrees are determined by the set pair analysis (SPA) theory. According to the principle of scientificalness, representative, completeness and operability, the index systems and standard of water resources vulnerability evaluation are established based on the analysis of sensibility and adaptability which include five subsystems: climate change, water resources change, social and economic infrastructure, water use level and water security capability. The AHPSPA model is used to assess water resource vulnerability in Beijing with 26 indexes under eight kinds of future climate change scenarios. Certain and uncertain information quantity of the WRV is calculated by connection numbers in the AHPSPA model. Results show that the WRV of Beijing is in the middle vulnerability (3 or III) under above-mentioned different climate change scenarios. The uncertain information is between 37.77 and 39.99 % in the WRV evaluation system in Beijing. Compared with present situation, the WRV will become better under scenario I and III and will become worse under scenario II, scenario IV, scenario representative concentration pathways (RCP)2.6, scenario RCP4.5, scenario RCP6.0 and scenario RCP8.5. In addition, we find that water resources change and water use level factors play more important role in the evaluation system of water resource vulnerability in Beijing. Finally, we make some suggestions for water resources management of Beijing.

Communicating the uncertainty in estimated greenhouse gas emissions from agriculture

In an effort to mitigate anthropogenic effects on the global climate system, industrialised countries are required to quantify and report, for various economic sectors, the annual emissions of greenhouse gases from their several sources and the absorption of the same in different sinks. These estimates are uncertain, and this uncertainty must be communicated effectively, if government bodies, research scientists or members of the public are to draw sound conclusions. Our interest is in communicating the uncertainty in estimates of greenhouse gas emissions from agriculture to those who might directly use the results from the inventory. We tested six methods of communication. These were: a verbal scale using the IPCC calibrated phrases such as ‘likely’ and ‘very unlikely’; probabilities that emissions are within a defined range of values; confidence intervals for the expected value; histograms; box plots; and shaded arrays that depict the probability density of the uncertain quantity. In a formal trial we used these methods to communicate uncertainty about four specific inferences about greenhouse gas emissions in the UK. Sixty four individuals who use results from the greenhouse gas inventory professionally participated in the trial, and we tested how effectively the uncertainty about these inferences was communicated by means of a questionnaire. Our results showed differences in the efficacy of the methods of communication, and interactions with the nature of the target audience. We found that, although the verbal scale was thought to be a good method of communication it did not convey enough information and was open to misinterpretation. Shaded arrays were similarly criticised for being open to misinterpretation, but proved to give the best impression of uncertainty when participants were asked to interpret results from the greenhouse gas inventory. Box plots were most favoured by our participants largely because they were particularly favoured by those who worked in research or had a stronger mathematical background. We propose a combination of methods should be used to convey uncertainty in emissions and that this combination should be tailored to the professional group.

Proper Scoring Rules That Are Sensitive to Bracketing

From forecasting competitions to conditional value-at-risk requirements, the use of multiple quantile assessments is growing in practice. To evaluate them, we characterize the general class of proper scoring rules for a forecaster's multiple quantiles of a single uncertain quantity of interest. The general rule is additive in the component scores. Each component contains a function that measures its quantile's distance from the realization and weights its contribution to the overall score. To determine this function, we propose that the score of a group's combined quantile should be better than that of a randomly-selected forecaster's quantile only when the forecasters bracket the realization, or their quantiles do not all fall on the same side of the realization. If a score satisfies this property, we say it is sensitive to bracketing. We characterize the class of proper scoring rules that is sensitive to bracketing when the decision maker uses a generalized average to combine forecasters' quantiles. Finally, we show how weights can be set to match the payoffs in many important business contexts.

Generalized Discrete Fuzzy Number and Application in Risk Evaluation

In this paper, the definition of generalized discrete fuzzy numbers is introduced, and a representation theorem of such fuzzy numbers is obtained. Based on the representation theorem, it is shown that the usual addition and multiplication which are defined by Zadeh’s extension principle do not preserve the closeness of the operation, and a new addition operation and a new multiplication operation are defined, which not only preserve the closeness of the operation, but also are easy to calculate. Then some weak orders on the generalized discrete fuzzy number space are defined, and their properties are investigated. At last, a practical example is given to show the application of the algorithmic version to rank uncertain or imprecise discrete quantity.