Partitioned Multiobjective Risk Method Research Articles

Quantitative Resilience Assessment under a Tri-Stage Framework for Power Systems

The frequent occurrence of natural disasters and malicious attacks has exerted unprecedented disturbances on power systems, accounting for the extensive attention paid to power system resilience. Combined with the evolving nature of general disasters, this paper proposes resilience assessment approaches for power systems under a tri-stage framework. The pre-disaster toughness is proposed to quantify the robustness of power systems against potential disasters, where the thinking of area division and partitioned multi-objective risk method (PMRM) is introduced. In the case of information deficiency caused by disasters, the during-disaster resistance to disturbance is calculated to reflect the real-time system running state by state estimation (SE). The post-disaster restoration ability consists of response ability, restoration efficiency and restoration economy, which is evaluated by Sequential Monte-Carlo Simulation to simulate the system restoration process. Further, a synthetic metric system is presented to quantify the resilience performance of power systems from the above three aspects. The proposed approaches and framework are validated on the IEEE RTS 79 system, and helpful conclusions are drawn from extensive case studies.

Adaptive multiplayer approach for risk‐based decision‐making: 2006 Virginia Gubernatorial Inauguration

AbstractThis paper introduces a multiplayer framework to collaboratively characterize sources of system risk from multiple expert perspectives, integrating several tools for holistic risk identification and risk management elicitation: the Adaptive Multiplayer Hierarchical Holographic Model (AMP‐HHM), the Risk Filtering, Ranking, and Management methodology, and the Partitioned Multiobjective Risk Method. Applicable to a broad set of systems engineering risk applications, the framework is illustrated with a case study focusing on risk assessment and management for the 2006 Gubernatorial Inauguration held in Williamsburg, VA, for which 272 different areas of risk were identified by collaborative means. Management options for three scenarios were defined, and their impacts on the inauguration were evaluated through expected and upper‐tail risk measures. As a final step, the analysis determined a set of noninferior options that can be used by decision‐makers to craft security strategies. A computer tool was developed to assist in the AMP‐HHM process, demonstrating the efficacy of the framework in collaborative work without synchronicity and regardless of geography. While the framework has been applied to a defense risk analysis in this paper, it can easily be applied to any problem where experts from disparate backgrounds, expertise, or geographies are working together to solve an unstructured, multidimensional problem. © 2011 Wiley Periodicals, Inc.

Partitioned multiobjective risk modeling of carcinogenic compounds in groundwater

Quantifying human cancer risk arising from exposure to contaminated groundwater is complicated by the many hydrogeological, environmental, and toxicological uncertainties involved. In this study, we used Monte Carlo simulation to estimate cancer risk associated with tetrachloroethene (PCE) dissolved in groundwater by linking three separate models for: (1) reactive contaminant transport; (2) human exposure pathways; and (3) the PCE cancer potency factor. The hydrogeologic model incorporates an analytical solution for a one-dimensional advective–dispersive–reactive transport equation to determine the PCE concentration in a water supply well located at a fixed distance from a continuous source. The pathway model incorporates PCE exposure through ingestion, inhalation, and dermal contact. The toxicological model combines epidemiological data from eight rodent bioassays of PCE exposure in the form of a composite cumulative distribution frequency curve for the human PCE cancer potency factor. We assessed the relative importance of individual model variables through their correlation with expected cancer risk calculated in an ensemble of Monte Carlo simulations with 20,000 trials. For the scenarios evaluated, three factors were most highly correlated with cancer risk: (1) the microbiological decay constant for PCE in groundwater, (2) the linear groundwater pore velocity, and (3) the cancer potency factor. We then extended our analysis beyond conventional expected value risk assessment using the partitioned multiobjective risk method (PMRM) to generate expected-value functions conditional to a 1 in 100,000 increased cancer risk threshold. This approach accounts for low probability/high impact outcomes separately from the conventional unconditional expected values. Thus, information on potential worst-case outcomes can be quantified for decision makers. Using PMRM, we evaluated the cost-benefit relationship of implementing several postulated risk management alternatives intended to mitigate the expected and conditional cancer risk. Our results emphasize the importance of hydrogeologic models in risk assessment, but also illustrate the importance of integrating environmental and toxicological uncertainty. When coupled with the PMRM, models integrating uncertainty in transport, exposure, and potency constitute an effective risk assessment tool for use within a risk-based corrective action (RBCA) framework.

Applying the partitioned multiobjective risk method (PMRM) to portfolio selection.

The analysis of risk-return tradeoffs and their practical applications to portfolio analysis paved the way for Modern Portfolio Theory (MPT), which won Harry Markowitz a 1992 Nobel Prize in Economics. A typical approach in measuring a portfolio's expected return is based on the historical returns of the assets included in a portfolio. On the other hand, portfolio risk is usually measured using volatility, which is derived from the historical variance-covariance relationships among the portfolio assets. This article focuses on assessing portfolio risk, with emphasis on extreme risks. To date, volatility is a major measure of risk owing to its simplicity and validity for relatively small asset price fluctuations. Volatility is a justified measure for stable market performance, but it is weak in addressing portfolio risk under aberrant market fluctuations. Extreme market crashes such as that on October 19, 1987 ("Black Monday") and catastrophic events such as the terrorist attack of September 11, 2001 that led to a four-day suspension of trading on the New York Stock Exchange (NYSE) are a few examples where measuring risk via volatility can lead to inaccurate predictions. Thus, there is a need for a more robust metric of risk. By invoking the principles of the extreme-risk-analysis method through the partitioned multiobjective risk method (PMRM), this article contributes to the modeling of extreme risks in portfolio performance. A measure of an extreme portfolio risk, denoted by f(4), is defined as the conditional expectation for a lower-tail region of the distribution of the possible portfolio returns. This article presents a multiobjective problem formulation consisting of optimizing expected return and f(4), whose solution is determined using Evolver-a software that implements a genetic algorithm. Under business-as-usual market scenarios, the results of the proposed PMRM portfolio selection model are found to be compatible with those of the volatility-based model. However, under extremely unfavorable market conditions, results indicate that f(4) can be a more valid measure of risk than volatility.

Evaluation and Presentation of Dam Failure and Flood Risks

Safety studies for existing dams have found that some do not satisfy current estimates of the probable maximum flood (PMF). An event or influence diagram can describe the random factors that contribute to major inflow floods and that determine reservoir operation and possible downstream damages during a flood event. This allows calculation of the probability of dam failure and the distributions of damages and loss of life using combinations of various analytical and Monte Carlo methods. This paper discusses the efficiency of different evaluation methods: event trees, simple Monte Carlo sampling, Latin hypercube sampling, importance sampling, and a analytical/stratified Monte Carlo (A/SMC) method. The analysis suggests that the A/SMC method and importance sampling have great potential for the efficient estimation of dam failure risks. Numerical examples employ the distributions of damages and loss of life to show the character of trade-offs presented by many dam safety decisions and illustrate problems with the partitioned multiobjective risk method (PMRM). The use of partial expected damage and loss of life functions is recommended to show the importance of low-probability/high-consequence events.

RISK OF EXTREME EVENTS IN A MULTIOBJECTIVE FRAMEWORK

: This paper summarizes advances made in risk-based decisionmaking in water resources through use of the partitioned multiobjective risk method (PMRM). (Risk is a measure of the probability and severity of adverse effects.) In the PMRM, the risk of extreme events is differentiated from risk involving less extreme damage severity and is evaluated within a multiobjective framework. Study of the extreme-event risk function f4(*) has addressed the following issues: methods for calculating f4(*); the sensitivity of f4(*) to various parameters, particularly to the partitioning point of the extreme-event range and the selection of probability density functions; insight provided by the statistics of extremes; and the impact of f4(*) on risk management, for example, in the application of the PMRM to water resources problems. In particular, this paper shares with the reader recent research results on the PMRM, the relationship between the statistics of extremes and the conditional expected value, derived formulas for f4(*), distribution-free estimates of f4(*), documented case studies in dam safety, and future research directions.

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.

Risk Assessment of Extreme Events: Application

The safety of many existing dams could be improved by modifying hem structurally in accordance with recent advances in statistical hydrology and mproved availability of meteorological and hydrological data. Any increase insafety that might be gained by structural changes must be balanced against their costs. The risk analysis methodology known as the partitioned multiobjective risk method (PMRM) is explored in this paper through a dam‐safety problem. The PMRM is well suited to the task of solving the probabilistic optimization problem posed by such risk‐versus‐cost considerations. With the PMRM, a number of conditional expected‐damage functions are generated. Of these, one that represents events of a more extreme and catastrophic character is of particular interest here. The close relationship that exists between the expectation of damage and the statistics of extremes is shown to simplify the implementation of the PMRM, and the relationship also permits the derivation of closed‐form equations that determine (for any partitioning of the probability axis) the expected damage, given that a flood with a return period that equals or exceeds n years occurs. Finally, an analysis is made as to how the choice of the distribution function representing the annual flood peaks might affect the conditional expectations.

Risk‐based analysis of extreme events

Mathematical expectation has traditionally been used in solving risk‐based decision making problems. However, this concept is not appropriate for decision making that affects public policy because it conceals extremes by commensurating events of different magnitudes and probabilities of occurrence. If a decision maker instead uses an expected value function in addition to a conditional expectation that captures extreme and catastrophic events, he will gain a more meaningful and more encompassing representation of the trade‐offs at hand. A theory relating conditional expectation to the statistics of extremes is developed in this paper. This expectation can now be viewed as the conditional expected risk given the occurrence of an event with a return period that equals or exceeds n years. The theory highlights the importance of using both the conditional and the unconditional expected risk in decision making. This fact has previously been recognized in the partitioned multi‐objective risk method (PMRM), a risk analysis methodology based on the concept of conditional expectation. The theory proposed in this paper provides a formulation for analyzing the sensitivity of subjectively chosen parameters in the PMRM. This theory will also simplify the practical implementation of the method.

Probability distributions and their partitioning

The partitioned multi‐objective risk method (PMRM) was developed for solving risk‐based multi‐objective decision making problems. Based on the premise that the expected value concept is not sufficient for proper decision making, the PMRM generates a number of conditional expected value functions (or risk functions) by partitioning the probability axis into probability ranges. The goal of partitioning the probability axis is to have better information on extreme events for decision making purposes. These conditional expectations are dependent on the chosen partitioning points. This paper analyzes how conditional expectations are sensitive to variations in partitioning. One of the risk functions is a measure of extreme and catastrophic events. By using the relationship between this particular risk function and the statistics of extremes, the sensitivity analysis is simplified. In many practical applications, it is difficult to determine which type of distribution function best represents the random process. Conditional expectations also depend on the choice of distribution, and the impact of this selection is discussed.

Multiobjective Risk‐Impact Analysis Method

This paper integrates two methodologies of risk and impact analysis—the partitioned multiobjective risk method (PMRM) and the multiobjective, multistage impact analysis method (MMIAM). The PMRM, in an effort to overcome the difficulties created by commensurating extreme events that have catastrophic impacts with those that have more frequent but less harmful consequences, introduces the use of conditional expectations for different regimes of probabilities and damages. The MMIAM is a multiobjective decisionmaking method for impact analysis which explicitly develops trade‐offs among different objectives at different stages or periods. Decision makers who must act under conditions of extreme risk and uncertainty often find that they are more interested in knowing “what not to do” than they are in optimizing their current objectives (the more traditional aim of decision‐making methodologies). This paper focuses on a way to answer their needs, in which major elements of both the PMRM and the MMIAM are combined in a new, integrated methodology, termed here the multiobjective risk‐impact analysis method (MRIAM). The new methodology incorporates risk and impact analysis within a dynamic multiobjective decision‐making framework. To demonstrate the usefulness of the integrated methodology, an example problem concerning the environmental effects of pollutant emissions over a number of years is formulated, solved, and analyzed.

Integrated risk and uncertainty assessment in water resources within a multiobjective framework

Abstract The process of risk-and-uncertainty assessment in water resources is examined within a multiobjective framework. In this context, appropriate methodologies — the surrogate worth trade-off (SWT) method, the multiobjective statistical method (MSM), the partitioned multiobjective risk method (PMRM), the risk dispersion index method (RDIM), and the uncertainty sensitivity index method (USIM) — are very briefly discussed. A need for an approach that integrates the assessment of risk and uncertainty within a multiobjective framework, accounting for random variables with both known and unknown probability distributions is suggested. This paper is dedicated to the memory of Ven Te Chow, a distinguished educator, prominent researcher, renowned scholar and dedicated public servant, who dominantly influenced the field of hydrology and water resources.