Risk of Extreme Events in Multiobjective Decision Trees Part 2. Rare Events

Earlier work with decision trees identified nonseparability as an obstacle to minimizing the conditional expected value, a measure of the risk of extreme events, by the well-known method of averaging out and folding back. This second of two companion papers addresses the conditional expected value that is defined as the expected outcome assuming that a random variable is observed only in the upper 100 (1 −α) percent of potential outcomes, where α is a cumulative probability preselected by the decision maker. An approach is proposed to overcome the need to evaluate all policies in order to identify the optimal policy. The approach is based in part on approximating the conditional expected value by using statistics of extremes. An existing convenient approximation of the conditional expected value is shown to be separable into two constituent elements of risk and can thus be optimized, along with other objectives including the unconditional expected value of the outcome, in a multiobjective decision tree. An example of sequential decision making for remediation or environmental contamination is provided. The importance of the results for risk analyis beyond the minimization of conditional expected values is pointed out.     

Multiobjective Decision-Tree Analysis1

Single-objective-based decision-tree analysis has been extensively and successfully used in numerous decision-making problems since its formal introduction by Howard Raiffa more than two decades ago. This paper extends the traditional methodology to incorporate multiple noncommensurate objective functions and use of the conditional expected value of the risk of extreme and catastrophic events. The proposed methodology considers the cases where (a) a finite number of actions are available at each decision node and (b) discrete or continuous states of nature can be presented at each chance node. The proposed extension of decision-tree analysis is introduced through an example problem that leads the reader step-by-step into the methodological procedure. The example problem builds on flood warning systems. Two noncommensurate objectives—the loss of lives and the loss of property (including monetary costs of the flood warning system)–are incorporated into the decision tree. In addition, two risk measures—the common expected value and the conditional expected value of extreme and catastrophic events—are quantified and are also incorporated into the decision-making process. Theoretical difficulties associated with the stage-wise calculation of conditional expected values are identified and certain simplifying assumptions are made for computational tractibility. In particular, it is revealed that decisions concerning experimentation have a very interesting impact on the noninferior solution set of options—a phenomenon that has no equivalence in the single-objective case.     

Selection of Probability Distributions in Characterizing Risk of Extreme Events

Use of probability distributions by regulatory agencies often focuses on the extreme events and scenarios that correspond to the tail of probability distributions. This paper makes the case that assessment of the tail of the distribution can and often should be performed separately from assessment of the central values. Factors to consider when developing distributions that account for tail behavior include (a) the availability of data, (b) characteristics of the tail of the distribution, and (c) the value of additional information in assessment. The integration of these elements will improve the modeling of extreme events by the tail of distributions, thereby providing policy makers with critical information on the risk of extreme events. Two examples provide insight into the theme of the paper. The first demonstrates the need for a parallel analysis that separates the extreme events from the central values. The second shows a link between the selection of the tail distribution and a decision criterion. In addition, the phenomenon of breaking records in time-series data gives insight to the information that characterizes extreme values. One methodology for treating risk of extreme events explicitly adopts the conditional expected value as a measure of risk. Theoretical results concerning this measure are given to clarify some of the concepts of the risk of extreme events.     

Risk-Adjusted Mission Value: Trading Off Mission Risk for Mission Value

This article develops a decision-theoretic methodology for the risk-adjusted mission value (RAMV) for selecting between alternative missions in the presence of uncertainty in the outcomes of the missions. This methodology permits trading off mission risk for mission value, something that probabilistic risk analysis cannot do unless it explicitly incorporates both mission value and risk aversion of the project management. The methodology, in its complete implementation, is consistent with the decision theory known as expected utility theory, although it differs from conventional decision theory in that the probabilities and all but one of the utilities are not those of the decision maker. The article also introduces a new interpretation of risk aversion. The methodology is consistent with the elementary management concept concerning division of labor. An example is presented for selecting between discrete alternatives-four landing sites on Mars. A second example is presented for selecting among a set of continuous alternatives-a comet flyby distance. The methodology is developed within the context of scientific missions, but the methodology is equally applicable to any situation requiring outcome value judgments, probability judgments, and risk aversion judgments by different constituencies.     

Strategic supply network planning with vendor selection under consideration of risk and demand uncertainty

We present a stochastic version of a three-layer supply network planning problem that includes the selection of vendors that must be equipped with company-specific tools. The configuration of a supply network must be determined by using demand forecasts for a long planning horizon to meet a given service level. The risk induced by the uncertain demand is explicitly considered by incorporating the conditional value at risk. The objective is to maximize the weighted sum of the expected net present value of discounted cash flows and the conditional value at risk. This would lead to a non-linear model formulation that is approximated by a mixed-integer linear model. This approximation is realized by a piecewise linearization of the expected backlogs and physical inventory as non-linear functions of cumulative production quantities. A two-stage stochastic programming approach is proposed. Our numerical analysis of generic test instances indicates that solving the linearized model formulation yields a robust and stable supply network configuration when demand is uncertain.     

基于CVaR约束的多产品订货风险决策模型

过去随机环境下多产品订货往往以期望值作为唯一决策准则,没有将风险控制纳入决策范畴,与实际决策过程不相符合.为给具有不同风险偏好的决策者提供合适的决策分析工具,文章在分析投资组合和产品组合存在某种相似性的基础上,借鉴金融工程领域广泛应用的条件风险值方法,建立多产品最优订货决策模型,并对模型进行了检验,发现它完全符合决策者的直觉要求.而且,由于所建的模型最终可以表示为一个线性规划问题,因此即使是大规模的产品组合问题也可以借助工具软件求解.     

Evaluating airline competitiveness using multiattribute decision making

This paper presents an objective approach to the evaluation of airline competitiveness. The evaluation problem is formulated as a multiattribute decision making model and solved by three widely used methods (the simple additive weighting method, the weighted product method and the technique for order preference by similarity to ideal solution) based on multiattribute value theory. A new empirical validation procedure is developed to deal with the inconsistency problem of evaluation outcomes produced by the three methods. The procedure selects the evaluation outcome which has a minimum expected value loss. An empirical study on Taiwan's five major domestic airlines is conducted to demonstrate the effectiveness of the approach. To measure and compare overall competitiveness of the airlines, five competitiveness dimensions and their associated objective performance measures on both efficiency and effectiveness are identified. The result of empirical validation for the three methods suggests the use of the simple additive weighting method. The evaluation outcome helps an airline identify its competitive advantages relative to its competitors. The objective approach presented is particularly applicable when subjective judgements on performance ratings and attribute weights are not reliable, or suitable decision makers are not available.     

Generalized Quantification of Risk Associated with Extreme Events1

In the partitioned multiobjective risk method (PMRM) the probability axis is typically partitioned into three regimes: high-exceedance low-consequence, intermediate-exceedance intermediate-consequence, and low-exceedance high-consequence (LE/HC). For each regime, the PMRM generates a conditional expected risk-function given that the damage lies within the regime. The theme of this paper is the conditional expected-risk function for the LE/HC regime. This function, denoted by f₄(.), captures the behavior of the “extreme events” of an underlying decision-making problem. The PMRM offers two advantages: (a) it isolates LE/HC events, allowing the decision-maker(s) to focus on the impacts of catastrophies; and (b) it generates more valuable information than that obtained from the common unconditional expected-risk function. Theoretical problems may arise from uncertainty about the behavior of the tail of the risk curve describing the underlying frequency of damages. When the number of physical observations in small (e.g., in flood frequency analysis), the analyst is forced to make assumptions about the density of damages. Each succeeding distributional assumption will generate a different value of f₄(.). An added dimension of difficulty is also created by the sensitivity of f₄(.) to the choice of the boundary of the LE/HC regime. This paper has two overall objectives: (a) to present distribution-free results concerning the magnitude of f₄(.); and (b) to use those results to obtain a distribution-free estimate of the sensitivity of f₄(.) to the choice of the boundary of the LE/HC regime. The above objectives are realized by extending, and further developing, existing inequalities for continuously distributed random variables.     

Selection of supply portfolio under disruption risks

This paper deals with the optimal selection of supply portfolio in a make-to-order environment in the presence of supply chain disruption risks. Given a set of customer orders for products, the decision maker needs to decide from which supplier to purchase custom parts required for each customer order to minimize total cost and mitigate the impact of disruption risks. The selection of suppliers and allocation of orders is based on price and quality of purchased parts and reliability of delivery. The two types of disruption scenarios are considered: scenarios with independent local disruptions of each supplier and scenarios with local and global disruptions that may result in all suppliers disruption simultaneously. The problem is formulated as a single- or bi-objective mixed integer program and a value-at-risk and conditional value-at-risk approach is applied to control the risk of supply disruptions. The proposed portfolio approach is capable of optimizing the supply portfolio by calculating value-at-risk of cost per part and minimizing expected worst-case cost per part simultaneously. Numerical examples are presented and some computational results are reported.     

COMPARATIVE EVALUATION OF INFORMATION UNDER TWO BUSINESS OBJECTIVES

Information economics models evaluate the value of information under the assumption that decision makers wish to maximize their expected payoff. This assumption has been criticized for not being realistic enough since decision makers usually consider more than one business objective and might be satisficers rather than optimizers. This paper attempts to apply an information economics model to decision situations where two business performance criteria, expected payoff and risk, are considered. In order to overcome the difficulty of unknown trade-off between the two criteria, one criterion is used as an objective to be optimized, while the other is set as a constraint. This may be interpreted as a combination of optimizing and satisficing approaches. It is shown how an information system can be evaluated in terms of both expected payoff and risk. The model suggests the trade-off between the two criteria as an additional trait of an information system. In the last part of the paper, a numerical example illustrates how a comparative evaluation of information structures is performed when risk minimization and expected payoff maximization are concurrently used as business performance criteria.     

The Risk Management of Third Parties During Construction in Multifunctional Urban Locations

Buildings above roads, railways, and existing buildings themselves are examples of multifunctional urban locations. The construction stage of those buildings is in general extremely complicated. Safety is one of the critical issues during the construction stage. Because the traffic on the infrastructure must continue during the construction of the building above the infrastructure, falling objects due to construction activities form a major hazard for third parties, i.e., people present on the infrastructure or beneath it, such as car drivers and passengers. This article outlines a systematic approach to conduct quantitative risk assessment (QRA) and risk management of falling elements for third parties during the construction stage of the building above the infrastructure in multifunctional urban locations. In order to set up a QRA model, quantifiable aspects influencing the risk for third parties were determined. Subsequently, the conditional probabilities of these aspects were estimated by historical data or engineering judgment. This was followed by integrating those conditional probabilities, now used as input parameters for the QRA, into a Bayesian network representing the relation and the conditional dependence between the quantified aspects. The outcome of the Bayesian network—the calculation of both the human and financial risk in quantitative terms—is compared with the risk acceptance criteria as far as possible. Furthermore, the effect of some safety measures were analyzed and optimized in relation with decision making. Finally, the possibility of integration of safety measures in the functional and structural building design above the infrastructure are explored.     

Estimating Conditional Probabilities of Terrorist Attacks: Modeling Adversaries with Uncertain Value Tradeoffs

Many analyses conducted to inform security decisions depend on estimates of the conditional probabilities of different attack alternatives. These probabilities are difficult to estimate since analysts have limited access to the adversary and limited knowledge of the adversary’s utility function, so subject matter experts often provide the estimates through direct elicitation. In this article, we describe a method of using uncertainty in utility function value tradeoffs to model the adversary’s decision process and solve for the conditional probabilities of different attacks in closed form. The conditional probabilities are suitable to be used as inputs to probabilistic risk assessments and other decision support techniques. The process we describe is an extension of value‐focused thinking and is broadly applicable, including in general business decision making. We demonstrate the use of this technique with simple examples.     

Optimizing the conditional value-at-risk in revenue management

Many service industries use revenue management to balance demand and capacity. The assumption of risk-neutrality lies at the heart of the classical approaches, which aim at maximizing expected revenue. In this paper, we give a comprehensive overview of the existing approaches, most of which were only recently developed, and discuss the need to take risk-averse decision makers into account. We then present a heuristic that maximizes conditional value-at-risk (CVaR). Although CVaR has become increasingly popular in finance and actuarial science due to its beneficial properties, this risk measure has not yet been considered in the context of revenue management. We are able to efficiently solve the optimization problem inherent in CVaR by taking advantage of specific structural properties that allow us to reformulate this optimization problem as a continuous knapsack problem. In order to demonstrate the applicability and robustness of our approach, we conduct a simulation study that shows that the new approach can significantly improve the risk profile in various scenarios.     

The Role of Quantitative Risk Assessments for Characterizing Risk and Uncertainty and Delineating Appropriate Risk Management Options, with Special Emphasis on Terrorism Risk

The purpose of this article is to discuss the role of quantitative risk assessments for characterizing risk and uncertainty and delineating appropriate risk management options. Our main concern is situations (risk problems) with large potential consequences, large uncertainties, and/or ambiguities (related to the relevance, meaning, and implications of the decision basis; or related to the values to be protected and the priorities to be made), in particular terrorism risk. We look into the scientific basis of the quantitative risk assessments and the boundaries of the assessments in such a context. Based on a risk perspective that defines risk as uncertainty about and severity of the consequences (or outcomes) of an activity with respect to something that humans value we advocate a broad risk assessment approach characterizing uncertainties beyond probabilities and expected values. Key features of this approach are qualitative uncertainty assessment and scenario building instruments.     

Risk of Extreme Events Under Nonstationary Conditions

The concept of the return period is widely used in the analysis of the risk of extreme events and in engineering design. For example, a levee can be designed to protect against the 100-year flood, the flood which on average occurs once in 100 years. Use of the return period typically assumes that the probability of occurrence of an extreme event in the current or any future year is the same. However, there is evidence that potential climate change may affect the probabilities of some extreme events such as floods and droughts. In turn, this would affect the level of protection provided by the current infrastructure. For an engineering project, the risk of an extreme event in a future year could greatly exceed the average annual risk over the design life of the project. An equivalent definition of the return period under stationary conditions is the expected waiting time before failure. This paper examines how this definition can be adapted to nonstationary conditions. Designers of flood control projects should be aware that alternative definitions of the return period imply different risk under nonstationary conditions. The statistics of extremes and extreme value distributions are useful to examine extreme event risk. This paper uses a Gumbel Type I distribution to model the probability of failure under nonstationary conditions. The probability of an extreme event under nonstationary conditions depends on the rate of change of the parameters of the underlying distribution.     

中国商品期货回报与现货价格变化测度研究——基于便利收益模型的视角

本文基于便利收益模型(CYM)的视角推导出商品期限结构、期货回报并对期货回报进行分解。选取我国三个商品期货交易所相关数据作为样本,对我国商品期货回报与现货价格变化进行测度研究。研究发现,在样本期内,商品期货回报和现货价格变化之间不存在密切关系;以展期收益或预期现货价格变化为条件的商品风险溢价具有时变性;平均展期收益反映了现货价格变化对风险溢价的预期偏离;期货期限结构、便利收益和展期收益准确地预测了现货价格变化。上述研究结果为我国商品期货回报与现货价格变化的测度和管理以及商品期货投资决策设计提供了一些有帮助的理论借鉴和操作性较强的方法选择。     

Evaluation of Two Approaches to Defining Extinction Risk under the U.S. Endangered Species Act

The predominant definition of extinction risk in conservation biology involves evaluating the cumulative distribution function (CDF) of extinction time at a particular point (the “time horizon”). Using the principles of decision theory, this article develops an alternative definition of extinction risk as the expected loss (EL) to society resulting from eventual extinction of a species. Distinct roles are identified for time preference and risk aversion. Ranges of tentative values for the parameters of the two approaches are proposed, and the performances of the two approaches are compared and contrasted for a small set of real‐world species with published extinction time distributions and a large set of hypothetical extinction time distributions. Potential issues with each approach are evaluated, and the EL approach is recommended as the better of the two. The CDF approach suffers from the fact that extinctions that occur at any time before the specified time horizon are weighted equally, while extinctions that occur beyond the specified time horizon receive no weight at all. It also suffers from the fact that the time horizon does not correspond to any natural phenomenon, and so is impossible to specify nonarbitrarily; yet the results can depend critically on the specified value. In contrast, the EL approach has the advantage of weighting extinction time continuously, with no artificial time horizon, and the parameters of the approach (the rates of time preference and risk aversion) do correspond to natural phenomena, and so can be specified nonarbitrarily.     

考虑背景风险的项目投资决策

在项目投资过程中不但面临项目风险,同时还面临背景风险,且背景风险与项目风险之间往往存在着一定的相关性。文章在已有研究的基础上,针对考虑背景风险的项目投资决策问题,分析了风险之间的相关性及相关程度对投资决策的影响。首先讨论了加性背景风险和乘性背景风险单独存在时,背景风险与项目风险之间的相关性对投资决策的影响;其次构建了两种背景风险同时存在情形下的投资模型,进而通过蒙特卡罗仿真方法给出不同相关程度下的仿真结果,在此基础上分析两种背景风险与项目风险之间的相关性及相关程度对投资决策的影响并给出相关研究结论。     

Economic Evaluation of Scale Dependent Technology Investments

We study the effect of financial risk on the economic evaluation of a project with capacity decisions. Capacity decisions have an important effect on the project̂s value through the up‐front investment, the associated operating cost, and constraints on output. However, increased scale also affects the financial risk of the project through its effect on the operating leverage of the investment. Although it has long been recognized in the finance literature that operating leverage affects project risk, this result has not been incorporated in the operations management literature when evaluating projects. We study the decision problem of a firm that must choose project scale. Future cash flow uncertainty is introduced by uncertain future market prices. The firm's capacity decision affects the firm's potential sales, its expected price for output, and its costs. We study the firm's profit maximizing scale decision using the CAPM model for risk adjustment. Our results include that project risk, as measured by the required rate of return, is related to the inverse of the expected profit per unit sold. We also show that project risk is related to the scale choice. In contrast, in traditional discounted cash flow analysis (DCF), a fixed prescribed rate is used to evaluate the project and choose its scale. When a fixed rate is used with DCF, a manager will ignore the effect of scale on risk and choose suboptimal capacity that reduces project value. S/he will also misestimate project value. Use of DCF for choosing scale is studied for two special cases. It is shown that if the manager is directed to use a prescribed discount rate that induces the optimal scale decision, then the manager will greatly undervalue the project. In contrast, if the discount rate is set to the risk of the optimally‐scaled project, the manager will undersize the project by a small amount, and slightly undervalue the project with the economic impact of the error being small. These results underline the importance of understanding the source of financial risk in projects where risk is endogenous to the project design.     

不确定环境下考虑风险厌恶的生产-分销网络优化模型及试验设计

考虑上游生产和下游需求不确定性，研究了由工厂、分销中心及终端市场构成的生产-分销网络优化设计问题。针对上游生产不确定性，考虑产生故障和无故障两种状态；针对下游市场需求不确定性，考虑其具有低、中和高三种状态。由于生产发生故障可能导致不合格品的产生，进一步考虑了在上游生产环节是否实施产品监测问题。综合网络运作成本和由不确定性导致的绩效风险，建立了由风险厌恶水平和悲观系数刻画的基于均值-条件风险值（CVaR）准则的生产-分销网络两阶段随机规划模型。特别地，针对由网络潜在节点数众多所导致的不确定情景规模过大的问题，采用情景缩减技术进行了情景筛选，降低了所建模型的求解难度。最后，进行了数值计算，分析了相关参数对网络运作绩效的影响，并给出了期望成本和条件风险值两个目标权衡的帕累托有效前沿。进一步，通过回归试验设计检验了决策者风险厌恶水平和悲观系数对所设计的生产-分销网络绩效的影响程度。结果表明，相对于决策者的风险厌恶程度，悲观系数对网络运作绩效的影响更大。