Approximate weighting method for multiattribute decision problems with imprecise parameters

A multiattribute decision problem with imprecise parameters refers to one in which at least one of the parameters such as attribute weights and value scores is not represented by precise numerical values. Some well-known types of incomplete attribute weights are chosen and analyzed to find their extreme points. In doing so, we show that their coefficients matrix, by itself or by the change of variables, belongs to a class of M-matrix which enables us to find its extreme points readily due to the inverse-positive property.The knowledge of extreme points not only helps us to prioritize alternatives but also supports iterative exploration of decision-maker’s preference by investigating modified extreme points caused by additional preference information. A wide range of eligible attribute weights, however, often fail to result in the best alternative or a complete ranking of alternatives. To address this situation, we consider an approximate weighting method, so called the minimizing squared deviations from extreme points (MSD) which locates the attribute weights at the barycenter of a weight set. Accordingly, the MSD approach extends the rank order centroid (ROC) weighting method which is known to outperform other approximate weighting methods in case of ranked attribute weights. The evidence of the MSD’s superiority over a linear program-based weighting method is verified via simulation analysis under different forms of incomplete attribute weights.     

考虑决策者心理行为的混合型随机多属性决策方法

针对混合型随机多属性决策问题,提出一种考虑决策者心理行为的决策分析方法。在该方法中,首先将具有离散型随机变量、灰色型随机变量和语言型变量形式的属性值规范化到区间内;然后将决策者给出的针对不同时期的属性期望视为参照点,并通过计算方案属性值与参照点的距离构建方案的益损矩阵;进一步地,依据累计前景理论,计算方案在不同属性上的收益和损失价值,并在此基础上,通过集结不同属性和不同时期的方案前景值确定各方案的综合前景值,进而依据得到的综合前景值确定方案排序结果。最后,通过一个算例说明了该方法的有效性和可行性。     

Establishing dominance between strategies with interval judgments of state probabilities

In decision-making under uncertainty, a decision-maker is required to specify, possibly with the help of decision analysts, point estimates of the probabilities of uncertain events. In this setting, it is often difficult to obtain very precise measurements of the decision-maker׳s probabilities on the states of nature particularly when little information is available to evaluate probabilities, available information is not specific enough, or we have to model the conflict case where several information sources are available.In this paper, imprecise probabilities are considered for representing the decision-maker׳s perception or past experience about the states of nature, to be specific, interval probabilities, which can be further categorized as (a) intervals of individual probabilities, (b) intervals of probability differences, and (c) intervals of ratio probabilities. We present a heuristic approach to modeling a wider range of types of probabilities as well as three types of interval probabilities. In particular, the intervals of ratio probabilities, which are widely used in the uncertain AHP context, are analyzed to find extreme points by the use of change of variables, not to mention the first two types of interval probabilities. Finally, we examine how these extreme points can be used to determine an ordering or partial ordering of the expected values of strategies.     

RUTA: A framework for assessing and selecting additive value functions on the basis of rank related requirements

In this paper, we present a new preference disaggregation method, called RUTA, which infers a set of additive value functions from the preference information referring to the desired ranks of some reference alternatives. Real-life experience indicates that people willingly refer to the range of allowed ranks that a particular alternative should attain, or to constraints on the final scores of the alternatives. We develop a mathematical model for incorporating such preference information via mixed-integer linear programming (MILP). Then, we discuss how decision making could be supported with the use of the already proposed extreme ranking analysis (ERA), which indicates the best and worst ranks gained by each alternative over the set of compatible preference model instances. We also introduce a new interactive UTA-like technique, which aims at selecting a single value function representing the outcomes of ERA. In the interactive process, the decision maker (DM) is assigning priorities to different pre-defined targets, which are built on results of ERA, and refer to the comparison of the best and/or worst ranks for pairs of alternatives. In particular, the DM may choose to emphasize or neglect the advantage of some alternatives over the others, in terms of results of ERA. In this way, one obtains a synthetic representation of extreme ranking analysis at a higher level of abstraction.     

A Survey of Approaches for Assessing and Managing the Risk of Extremes

In this paper, we review methods for assessing and managing the risk of extreme events, where extreme events are defined to be rare, severe, and outside the normal range of experience of the system in question. First, we discuss several systematic approaches for identifying possible extreme events. We then discuss some issues related to risk assessment of extreme events, including what type of output is needed (e.g., a single probability vs. a probability distribution), and alternatives to the probabilistic approach. Next, we present a number of probabilistic methods. These include: guidelines for eliciting informative probability distributions from experts; maximum entropy distributions; extreme value theory; other approaches for constructing prior distributions (such as reference or noninformative priors); the use of modeling and decomposition to estimate the probability (or distribution) of interest; and bounding methods. Finally, we briefly discuss several approaches for managing the risk of extreme events, and conclude with recommendations and directions for future research.     

Adjusting Triangular Distributions for Judgmental Bias

The common problem in risk analysis of correctly specifying a probability distribution about an estimate in situations when few data are available is examined. In the absence of data, experts are sometimes used to give a lowest and highest conceivable estimate. Triangular distributions are well suited for these situations when only a low, high, and most likely estimate are given. A problem, however, exists from the failure to adjust for biases when estimating extreme values. Various types of biases, which narrow the range of extreme estimates, are explored. A method is suggested for accounting for these biases by placing extreme estimates at specified percentile points rather than endpoints of a triangular distribution. Since most Monte Carlo models require end points of a triangular distribution, a closed-form expression for identifying the end points given two percentile points and a most likely point is derived. This method has been used extensively in developing cost risk estimates for the Ballistic Missile Defense Organization (BMDO).     

方案有不确定偏好的区间数相对熵群决策方法

研究了属性值是区间数并且已知方案偏好信息的多属性群决策问题。建立了每个方案客观偏好值与主观偏好值偏差的相对熵测度矩阵;基于客观信息和方案偏好信息的相对熵建立了属性权重模型;建立了一个新的区间数比较的可能度公式,基于可能度公式给出了方案排序方法,算例说明方法可行性。     

多属性决策决问题的满意度与赋权研究

多属性决策问题的决策中,决策者往往对属性上的数值存在一定的心理预期。首先,通过心理预期与实际数据获得决策对象在每个属性上的满意度,对决策对象进行筛选过滤;其次,提出属性值信息相容关系,利用属性值之间的相容度进行赋权,信息融合对满足决策者心理预期的决策对象排序择优;再次,提出决策对象满意度,并指出传统的排序方法获取的最优决策对象与决策者总体满意度最大的决策对象并不等价。具体算例表明,该方法科学有效且可行。     

基于区间灰色不确定语言的多准则决策方法

定义了区间灰色不确定语言及其运算法则和距离公式。拓展了连续区间数据的OWA(C-OWA)算子,提出了基于区间灰色不确定语言的有序加权C-OWA(IGULOWC-OWA)算子,并对其性质进行了分析。针对准则权重已知而方案的准则值为区间灰色不确定语言的多准则决策问题,提出了一种基于IGULOWC-OWA算子的多准则决策方法。该方法通过对各方案在各个准则下的评价值进行转换,将区间灰色不确定语言转化为不确定语言,然后,利用集结算子计算出各个方案的综合准则值,进而得出各方案的排序。最后通过算例表明了方法的可行性和有效性。     

基于区间灰色区间数的多准则决策方法

将灰色模糊数扩展,定义了区间灰色区间数及其距离.针对准则值为区间灰色区间数的信息不完全的多准则决策问题,提出了一种基于目标规划法的灰色模糊多准则决策方法,求得最优权重向量,从而得到各方案的排序.最后给出一个算例说明方法的可行性和有效性.     

PAHAP: A Pairwise Aggregated Hierarchical Analysis of Ratio-Scale Preferences

In this paper, we present a Pairwise Aggregated Hierarchical Analysis of Ratio-Scale Preferences (PAHAP), a new method for solving discrete alternative multicriteria decision problems. Following the Analytic Hierarchy Process (AHP), PAHAP uses pairwise preference judgments to assess the relative attractiveness of the alternatives. By first aggregating the pairwise judgment ratios of the alternatives across all criteria, and then synthesizing based on these aggregate measures, PAHAP determines overall ratio scale priorities and rankings of the alternatives which are not subject to rank reversal, provided that certain weak consistency requirements are satisfied. Hence, PAHAP can serve as a useful alternative to the original AHP if rank reversal is undesirable, for instance when the system is open and criterion scarcity does not affect the relative attractiveness of the alternatives. Moreover, the single matrix of pairwise aggregated ratings constructed in PAHAP provides useful insights into the decision maker's preference structure. PAHAP requires the same preference information as the original AHP (or, altematively, the same information as the Referenced AHP, if the criteria are compared based on average (total) value of the alternatives). As it is easier to implement and interpret than previously proposed variants of the conventional AHP which prevent rank reversal, PAHAP also appears attractive from a practitioner's viewpoint.     

Heuristics for prioritizing pair-wise elicitation questions with additive multi-attribute value models

Additive value models are widely used in Multiple Criteria Decision Analysis. Direct elicitation of the value model preference parameters can impose excessive cognitive burden on the decision maker. Indirect techniques that employ pair-wise questions have been proposed for lowering the elicitation effort. In all practically relevant problems, more than a single question needs to be answered for arriving at a sufficiently precise outcome. The selection and ordering of questions affects the number of answers required for ranking the decision alternatives. However, evaluating all possible questions and answers is intractable due to the search space being, in the worst case, of factorial size. This paper develops heuristics for prioritizing pair-wise elicitation questions based on (1) necessary preference relations, (2) extreme ranks attained by the alternatives, (3) pair-wise preference indices, and (4) rank acceptability indices. We also introduce three metrics for assessing quality of a question prioritization heuristic. Numerical results allow us to identify a subset of heuristics that score well on our metrics in a variety of problem settings. This conclusion was validated in a real-world experiment where 101 subjects answered pair-wise questions to rank 10 mobile phone packages evaluated in terms of four criteria.     

Power Enhancement in High‐Dimensional Cross‐Sectional Tests

We propose a novel technique to boost the power of testing a high‐dimensional vector H : θ = 0 against sparse alternatives where the null hypothesis is violated by only a few components. Existing tests based on quadratic forms such as the Wald statistic often suffer from low powers due to the accumulation of errors in estimating high‐dimensional parameters. More powerful tests for sparse alternatives such as thresholding and extreme value tests, on the other hand, require either stringent conditions or bootstrap to derive the null distribution and often suffer from size distortions due to the slow convergence. Based on a screening technique, we introduce a “power enhancement component,” which is zero under the null hypothesis with high probability, but diverges quickly under sparse alternatives. The proposed test statistic combines the power enhancement component with an asymptotically pivotal statistic, and strengthens the power under sparse alternatives. The null distribution does not require stringent regularity conditions, and is completely determined by that of the pivotal statistic. The proposed methods are then applied to testing the factor pricing models and validating the cross‐sectional independence in panel data models.     

Community Resilience and Decision Theory Challenges for Catastrophic Events

Extreme and catastrophic events pose challenges for normative models of risk management decision making. They invite development of new methods and principles to complement existing normative decision and risk analysis. Because such events are rare, it is difficult to learn about them from experience. They can prompt both too little concern before the fact, and too much after. Emotionally charged and vivid outcomes promote probability neglect and distort risk perceptions. Aversion to acting on uncertain probabilities saps precautionary action; moral hazard distorts incentives to take care; imperfect learning and social adaptation (e.g., herd‐following, group‐think) complicate forecasting and coordination of individual behaviors and undermine prediction, preparation, and insurance of catastrophic events. Such difficulties raise substantial challenges for normative decision theories prescribing how catastrophe risks should be managed. This article summarizes challenges for catastrophic hazards with uncertain or unpredictable frequencies and severities, hard‐to‐envision and incompletely described decision alternatives and consequences, and individual responses that influence each other. Conceptual models and examples clarify where and why new methods are needed to complement traditional normative decision theories for individuals and groups. For example, prospective and retrospective preferences for risk management alternatives may conflict; procedures for combining individual beliefs or preferences can produce collective decisions that no one favors; and individual choices or behaviors in preparing for possible disasters may have no equilibrium. Recent ideas for building “disaster‐resilient” communities can complement traditional normative decision theories, helping to meet the practical need for better ways to manage risks of extreme and catastrophic events.     

A deterministic model for the housing decision

The decision to rent or buy housing, or to choose among various housing acquisition alternatives, represents an important and complex financial problem, which confronts virtually all individuals. This paper develops a comprehensive model (within the institutional and legislative context of the United States) which provides theoretically appropriate criteria for evaluating personal housing alternatives. Empirical results clearly indicate the economic consequences of housing alternatives, and both substantiate some and contradict other “wisdoms” concerning the housing decision.     

Two-Level Hierarchical Transportation Networks: A New Set of Problems and Practical Applications*

This paper defines and solves a set of new problems for two-level hierarchical transportation networks. Given a set of points, the objective is to find the path of a primary link such that a weighted sum of the lengths of the perpendiculars from each point to the primary link and the distance between the feet of the two extreme perpendiculars is minimized. The perpendiculars are the connecting or secondary links and the primary link could be a straight line or a curve. The paper provides an algorithm and computational results for the case in which the primary link is a straight line. Implementation approaches are also discussed.     

THE INTERACTION OF TRANSPORTATION AND INVENTORY DECISIONS

The management of inventories and determination of transportation policy are normally carried out by different groups of people in an organization. These activities interact, however, when the transportation is used to replace inventory and when the transportation alternatives have different speed, reliability, and cost characteristics. This paper presents exact and heuristic procedures for jointly determining the inventory reorder points, order quantities, and transportation alternatives that provide minimum total transportation and inventory costs. The effectiveness of each procedure is demonstrated for a broad range of problems. The heuristic procedure was also applied to actual company data to determine the potential for cost savings. The results showed that savings were possible both from improving the current procedures and from applying the procedure developed in this study.     

三端点区间数互反判断矩阵的排序方法研究

本文研究了三端点区间数互反判断矩阵的一致性和排序方法。在三端点区间数互反判断矩阵完全一致性概念的基础上,首次将矩阵特征向量思想引入三端点区间数互反判断矩阵之中——研究了三端点区间数互反判断矩阵的一致性与权重向量之间的类似特征向量关系,并运用线性规划模型给出一种处理三端点区间数决策者对方案属性权重的方法,然后通过三端点区间数权重向量的期望值,进行方案集结排序。最后通过具体的案例,验证了所提出方法的有效性和适用性。     

权重信息完全未知的灰色多属性群决策方法研究

基于灰色系统理论的思想和方法,探讨了决策方案的属性值为区间灰数、属性权重及各决策人的权威权重完全未知的灰色多属性群决策问题。通过分析,引入了个体理想最优方案向量、理想专家、方案间优势度和方案集中方案间优势比较矩阵等概念。根据区间灰数的本质特征,定义了两区间灰数的新的相离度,构建了基于两区间灰数相离度的灰色区间关联系数公式及灰色关联度。建立了基于投影的特征向量法和模糊互补判断矩阵排序方法的两种决策算法,给出的算法避免了权重的计算。文中实例分析说明了所提出的灰色多属性群决策方法的合理性及其算法的有效性。     

Expanding the Reach of Participatory Risk Management: Testing an Online Decision‐Aiding Framework for Informing Internally Consistent Choices

This article presents research aimed at developing and testing an online, multistakeholder decision‐aiding framework for informing multiattribute risk management choices associated with energy development and climate change. The framework was designed to provide necessary background information and facilitate internally consistent choices, or choices that are in line with users’ prioritized objectives. In order to test different components of the decision‐aiding framework, a six‐part, 2 × 2 × 2 factorial experiment was conducted, yielding eight treatment scenarios. The three factors included: (1) whether or not users could construct their own alternatives; (2) the level of detail regarding the composition of alternatives users would evaluate; and (3) the way in which a final choice between users’ own constructed (or highest‐ranked) portfolio and an internally consistent portfolio was presented. Participants’ self‐reports revealed the framework was easy to use and providing an opportunity to develop one's own risk‐management alternatives (Factor 1) led to the highest knowledge gains. Empirical measures showed the internal consistency of users’ decisions across all treatments to be lower than expected and confirmed that providing information about alternatives’ composition (Factor 2) resulted in the least internally consistent choices. At the same time, those users who did not develop their own alternatives and were not shown detailed information about the composition of alternatives believed their choices to be the most internally consistent. These results raise concerns about how the amount of information provided and the ability to construct alternatives may inversely affect users’ real and perceived internal consistency.