Probabilistic dominance criteria for comparing uncertain alternatives: A tutorial

This paper is a tutorial which demonstrates the current state-of-the-art methods for incorporating risk into project selection decision making. The projects under consideration might be R&D, IT, or other capital expenditure programs. We will show six decision making methods: 1. mean-variance (MV), 2. mean-semivariance, 3. mean-critical probability, 4. stochastic dominance, 5. almost stochastic dominance (ASD), and 6. mean-Gini. We will also describe the assumptions about the risk attitudes of the decision maker which are associated with each of the techniques. While all these methods have been previously applied elsewhere, this is the first paper which shows all of their applications in the project selection context, together with their interrelationships, strengths and weaknesses. We have applied all six techniques to the same group of five hypothetical projects and evaluated the resulting nondominated sets. Among the methods reviewed here, stochastic dominance is recommended because it requires the least restrictive assumptions. ASD and mean-Gini are recommended when stochastic dominance is not practical or when it does not yield definitive choices. MV, mean-semivariance, and mean-critical probability are shown to be flawed.     

风险规避型供应链的收益共享机制研究

探讨了由风险规避型零售商与风险规避型供应商组成的两级供应链协作激励问题.首先研究了Stackelberg主从对策模型下风险规避型零售商与风险规避型供应商的决策行为,并分析了双方的风险规避系数对双方决策行为的影响.然后,研究了垂直一体化模型下风险规避型供应链的决策行为,同样分析了双方风险规避系数对供应链决策行为的影响程度.最后,提出采用收益共享契约机制来协调该供应链.并通过数值实验进一步分析供应商与零售商的风险规避系数对双方的决策行为及收益共享系数的影响程度.     

Stochastic Dominance: An Approach to Decision Making Under Risk

This paper introduces stochastic dominance as a technique to reduce the set of possible actions that a decision maker must consider in a decision problem under risk. The procedure usually does not choose an optimal action, but instead eliminates certain actions as unacceptable. Very little need be known about the decision maker's utility function. Two possible applications are presented: upgrading buildings to better withstand an earthquake; and choosing a site for a LNG facility.     

A STOCHASTIC DOMINANCE ORDERING OF SCHEDULING RULES

This paper applies stochastic dominance (SD) preference-ordering criteria to job shop scheduling rules. A simulation model of a hypothetical dual-constrained job shop is used to derive several measures of shop performance for a number of dispatching/due-date scheduling policies. The results presented suggest that previous research conclusions concerning the relative performance of dispatching scheduling rules may need to be reconsidered if production schedulers are risk-averse utility maximizers.     

A two-stage stochastic programming approach for identifying optimal postponement strategies in supply chains with uncertain demand

In the course of globalization, applying mass-customization strategies has led to a high diversity of variants in many economic sectors. Thus, customer demands are often less predictable, and handling increasing inventory stocks as well as avoiding shortfalls have become particularly important. All these complexity drivers result in higher supply chain risks. Postponement strategies have been proposed as a suitable approach to address these problems. Although the concept of postponement and its impact on the supply chain are theoretically well discussed, optimally configuring the entire production and distribution activities is still challenging. We present a two-stage stochastic mixed-integer linear programming model, which comprises an integrated production and distribution planning approach, and considers postponement concepts. In comparison to earlier approaches that examine postponement strategies, our model supports the decision maker under demand uncertainty and considers lead times, penalty costs for shortfalls, as well as inventory-keeping decisions over a tactical planning horizon. This allows an integrated investigation of both form and logistics postponement concepts. Moreover, we consider the decision maker’s risk attitude identifying non-dominated profitable and risk-averse strategies. We illustrate the benefits of the model by using a case study from the apparel industry, and present the results of a sensitivity analysis with respect to varying demand uncertainty and demand correlations as well as different preferences regarding risk aversion. Furthermore, we carry out performance and quality benchmarks and compare the results of a standard mixed-integer linear programming solver, a parallel nested Benders approach and a sample average approximation technique.     

Selecting a Portfolio of Technologies: An Application of Decision Analysis

The Department of Energy is faced with the critical and complex decision of selecting technologies for waste site remediation. This research focuses on developing a decision support tool which assists the decision maker to find an optimal portfolio. A portfolio consists of selecting the appropriate processes for a remediation site and, subsequently, selecting a technology for each process so that the decision maker's objectives are achieved. The measures supporting the objectives are risk, life-cycle cost, and time required for remediation. The model uses exponential attribute utility functions with an additive objective function. The model provides the decision maker with estimates of the cost and time distributions, and their associated utility. Cumulative frequency distributions illustrate the dominance of technology choices and the variance in the results. The model permits sensitivity analysis in the form of rainbow and tornado diagrams to display the effects of changes in the values of the input variables. Overall, the model provides a generic technology selection tool that can be used to make better informed decisions and may be easily manipulated to reflect changes in the remediation process.     

Decision Making with Uncertain Judgments: A Stochastic Formulation of the Analytic Hierarchy Process*

In the analytic hierarchy process (AHP), priorities are derived via a deterministic method, the eigenvalue decomposition. However, judgments may be subject to error. A stochastic characterization of the pairwise comparison judgment task is provided and statistical models are introduced for deriving the underlying priorities. Specifically, a weighted hierarchical multinomial logit model is used to obtain the priorities. Inference is then conducted from the Bayesian viewpoint using Markov chain Monte Carlo methods. The stochastic methods are found to give results that are congruent with those of the eigenvector method in matrices of different sizes and different levels of inconsistency. Moreover, inferential statements can be made about the priorities when the stochastic approach is adopted, and these statements may be of considerable value to a decision maker. The methods described are fully compatible with judgments from the standard version of AHP and can be used to construct a stochastic formulation of it.     

LEARNING WITH DETERMINISTIC DECISION RULES

While many problems of uncertainty are commonly analyzed by means of stochastic models, under certain circumstances this may not be an appropriate approach. The latter situation arises when the decision maker knows that the uncertain variables are not generated by a stochastic process, or when he is unwilling, or unable, to compute subjective probabilities. One of the nonstochastic approaches to uncertainty is the expectational approach in which the decision maker forms deterministic expectations about the uncertain aspects of his environment. This paper is concerned with some criteria for selecting among available expectations, or anticipations functions, and the possibility of ordering them according to these criteria. This study focuses especially on the learning criterion. The discussion brings out conceptual problems in connection with the definition of learning, as well as some technical difficulties that one encounters when attempting to compare different anticipations functions from the point of view of the learning criterion. As an illustration of the issues discussed, the paper reports on the results of some simulated decision rules. These show that decision rules in which no learning takes place, and in which some information is ignored, may perform better than more sophisticated rules.     

Decision making with stochastic dominance: An expository review

This paper surveys the use of stochastic dominance to decision making under uncertainty. The first part presents the relevant definitions and some properties of distributions satisfying one of the stochastic dominance conditions. These properties include restrictions on moments, an invariance property, and properties of random variables related by an exact formula.The second part contains some applications of the stochastic dominance method and especially the problem of selecting optimal portfolios. Most of the results in this section deal with conditions that make diversification an optimal strategy.     

Joint supplier selection and scheduling of customer orders under disruption risks: Single vs. dual sourcing

This paper presents a stochastic mixed integer programming approach to integrated supplier selection and customer order scheduling in the presence of supply chain disruption risks, for a single or dual sourcing strategy. The suppliers are assumed to be located in two different geographical regions: in the producer's region (domestic suppliers) and outside the producer's region (foreign suppliers). The supplies are subject to independent random local disruptions that are uniquely associated with a particular supplier and to random semi-global (regional) disruptions that may result in disruption of all suppliers in the same geographical region simultaneously. The domestic suppliers are relatively reliable but more expensive, while the foreign suppliers offer competitive prices, however material flows from these suppliers are more exposed to unexpected disruptions. Given a set of customer orders for products, the decision maker needs to decide which single supplier or which two different suppliers, one from each region, to select for purchasing parts required to complete the customer orders and how to schedule the orders over the planning horizon, to mitigate the impact of disruption risks. The problem objective is either to minimize total cost or to maximize customer service level. The obtained combinatorial stochastic optimization problem will be formulated as a mixed integer program with conditional value-at-risk as a risk measure. The risk-neutral and risk-averse solutions that optimize, respectively average and worst-case performance of a supply chain are compared for a single and dual sourcing strategy and for the two different objective functions. Numerical examples and computational results are presented and some managerial insights on the choice between the two sourcing strategies are reported.     

Stochastic Judgments in the AHP: The Measurement of Rank Reversal Probabilities

This paper presents a methodology for analyzing Analytic Hierarchy Process (AHP) rankings if the pairwise preference judgments are uncertain (stochastic). If the relative preference statements are represented by judgment intervals, rather than single values, then the rankings resulting from a traditional (deterministic) AHP analysis based on single judgment values may be reversed, and therefore incorrect. In the presence of stochastic judgments, the traditional AHP rankings may be stable or unstable, depending on the nature of the uncertainty. We develop multivariate statistical techniques to obtain both point estimates and confidence intervals of the rank reversal probabilities, and show how simulation experiments can be used as an effective and accurate tool for analyzing the stability of the preference rankings under uncertainty. If the rank reversal probability is low, then the rankings are stable and the decision maker can be confident that the AHP ranking is correct. However, if the likelihood of rank reversal is high, then the decision maker should interpret the AHP rankings cautiously, as there is a subtantial probability that these rankings are incorrect. High rank reversal probabilities indicate a need for exploring alternative problem formulations and methods of analysis. The information about the extent to which the ranking of the alternatives is sensitive to the stochastic nature of the pairwise judgments should be valuable information into the decision-making process, much like variability and confidence intervals are crucial tools for statistical inference. We provide simulation experiments and numerical examples to evaluate our method. Our analysis of rank reversal due to stochastic judgments is not related to previous research on rank reversal that focuses on mathematical properties inherent to the AHP methodology, for instance, the occurrence of rank reversal if a new alternative is added or an existing one is deleted.     

考虑随机需求与收入共享的风险规避型V2G备用决策模型

在CVaR风险度量准则下,构建了考虑随机需求与收入共享的风险规避型V2G备用决策模型,推导了集中和分散两种决策下渠道成员最优决策行为的解析解,并进一步比较分析了随机需求变量服从均匀分布时的均衡策略。研究发现,集中决策下的最优V2G备用预留因子与渠道整体的风险规避度正相关,而均衡时的V2G备用销售价格与渠道整体的风险规避度的相关性不确定,且受到随机需求变量的分布函数影响;分散决策下的最优V2G备用预留因子仅与电网公司的风险规避度有关,而均衡时的V2G备用销售价格受到电网公司的风险规避度、购电价格以及电动汽车用户收入共享系数等的共同影响;电动汽车用户的最优V2G备用收入共享系数与其风险规避度正相关,而与电网公司的风险规避度负相关。数值仿真结果表明,在绝大多数情形下收入共享合约并不能完美协调此类V2G备用渠道的分散决策行为。     

On qualitative multi-attribute group decision making and its consensus measure: A probability based perspective

This paper focuses on qualitative multi-attribute group decision making (MAGDM) with linguistic information in terms of single linguistic terms and/or flexible linguistic expressions. To do so, we propose a new linguistic decision rule based on the concepts of random preference and stochastic dominance, by a probability based interpretation of weight information. The importance weights and the concept of fuzzy majority are incorporated into both the multi-attribute and collective decision rule by the so-called weighted ordered weighted averaging operator with the input parameters expressed as probability distributions over a linguistic term set. Moreover, a probability based method is proposed to measure the consensus degree between individual and collective overall random preferences based on the concept of stochastic dominance, which also takes both the importance weights and the fuzzy majority into account. As such, our proposed approaches are based on the ordinal semantics of linguistic terms and voting statistics. By this, on one hand, the strict constraint of the uniform linguistic term set in linguistic decision making can be released; on the other hand, the difference and variation of individual opinions can be captured. The proposed approaches can deal with qualitative MAGDM with single linguistic terms and flexible linguistic expressions. Two application examples taken from the literature are used to illuminate the proposed techniques by comparisons with existing studies. The results show that our proposed approaches are comparable with existing studies.     

LINEAR PROGRAMMING WITH DISCRETE STOCHASTIC RESOURCES: AN ALTERNATIVE APPROACH ACCOUNTING FOR INFEASIBILITY*

In this paper an alternative to, or extension of, the chance-constrained method of stochastic programming is presented whereby an expected cost of infeasibility is included in the objective function. The problem is to select a solution to implement before the available resources are known where the adaption of a non-feasible solution to the resources available involves a system cost. While increasing the amount of computation required, the model enables the decision maker to more effectively trade off increased payoff for decreased likelihood of feasibility.     

On risk-averse maximum weighted subgraph problems

In this work, we consider a class of risk-averse maximum weighted subgraph problems (R-MWSP). Namely, assuming that each vertex of the graph is associated with a stochastic weight, such that the joint distribution is known, the goal is to obtain a subgraph of minimum risk satisfying a given hereditary property. We employ a stochastic programming framework that is based on the formalism of modern theory of risk measures in order to find minimum-risk hereditary structures in graphs with stochastic vertex weights. The introduced form of risk function for measuring the risk of subgraphs ensures that optimal solutions of R-MWS problems represent maximal subgraphs. A graph-based branch-and-bound (BnB) algorithm for solving the proposed problems is developed and illustrated on a special case of risk-averse maximum weighted clique problem. Numerical experiments on randomly generated Erdös-Rényi graphs demonstrate the computational performance of the developed BnB.     

考虑决策者后悔规避的风险投资项目选择方法

在风险投资项目选择过程中,决策者通常是有限理性的,会尽量避免选择可能会令其感到后悔的风险投资项目,而在涉及多个行业的风险投资项目选择过程中如何考虑决策者后悔规避的心理行为因素,这方面的研究特别需要关注。本文提出了一种涉及多个行业的考虑决策者后悔规避的风险投资项目选择方法。在该方法中,首先计算各风险投资项目在不同市场状态下的项目价值;然后,计算各组合市场状态发生的概率;进一步地,构建后悔函数刻画决策者后悔的心理感知,并通过计算不同行业的各风险投资项目的综合效用值得到风险投资项目的排序结果。最后,通过一个算例说明了本文提出方法的可行性与有效性。     

Relative Objective Functions in Multiobjective Decision Support Models*

This paper considers the importance of using the nondominated minimum in forming relative objective functions. Other minima can prevent the decision maker from reaching a preferred decision. They disturb the zero and distort the scale, thereby misguiding the decision maker by falsifying trade-off possibilities. Worse, some of these other minima can inadvertently hide whole regions of preferred solutions. Zeleny's multiple criteria simplex method is used to find the nondominated minimum for the linear case.     

风险厌恶零售商考虑信息预测成本的协调机制

在CVaR风险度量准则下,将信息预测成本作为运营成本之一,建立了风险厌恶的零售商的需求信息预测投入和订货量的联合决策随机模型,研究了风险态度对零售商最优决策的影响.通过分析得出在分散决策下,随着零售商风险厌恶程度的增加,零售商提高信息预测水平,以减小需求不确定风险.同时减少订货量,降低订货过量的风险.证明了在一定条件下,传统的收益共享契约能实现供应链协同,但适用范围较小.由信息预测成本分担与收益共享组成的联合契约可扩大供应链协同的范围,消除风险态度和双重边际对零售商决策行为的影响,使得零售商的需求预测水平和订货量同时达到系统风险中立环境下的最优水平.     

考虑损失规避零售商的多期多产品供应链网络均衡

供应链是一个具有层级结构的复杂网络,一般包括制造商层、零售商层和需求市场层等渠道成员。为描述供应链网络的动态特征,本文将决策时间离散划分为多个规划期,同一规划期内各类成员的决策环境相对稳定;相反在不同规划期之间可能发生改变。制造商生产多种类型的产品,零售商面临随机需求环境且为损失规避者。运用前景理论描述了零售商的损失规避行为,利用制造商的库存转移来描述相邻规划期间的关系。利用变分不等式、对偶理论和互补理论刻画了制造商层、零售商层和需求市场层的最优决策行为,并推导整个供应链网络的均衡条件。设计了求解模型的修正投影算法。通过算例阐明了零售商盈亏平衡点的特征,分析了损失规避系数对供应链网络企业最优决策的影响。研究表明:随着损失规避系数的增大,零售商的第1盈亏平衡点减小,第2盈亏平衡点增大;零售商的订货量减小,其期望利润和期望效用增加,相反制造商和消费者的利益均受损;随着缺货成本的增加,零售商需向制造商订购更多的产品来规避缺货损失,但这也同时增加了其过量订货和滞销的风险;当零售商在某一期的损失规避系数发生改变时,零售商和制造商需在整个规划期范围内调整策略。     

Impact of Number of Interactions,Different Interaction Patterns,and Human Inconsistencies on Some Hybrid Evolutionary Multiobjective Optimization Algorithms*

We investigate the impact of the number of human–computer interactions, different interaction patterns, and human inconsistencies in decision maker responses on the convergence of an interactive, evolutionary multiobjective algorithm recently developed by the authors. In our context “an interaction” means choosing the best and worst solutions among a sample of six solutions. By interaction patterns we refer to whether preference questioning is more front‐, center‐, rear‐, or edge‐loaded. As test problems we use two‐ to four‐objective knapsack problems, multicriteria scheduling problems, and multiobjective facility location problems. In the tests, two different preference functions are used to represent actual decision maker preferences, linear and Chebyshev. The results indicate that it is possible to obtain solutions that are very good or even nearly optimal with a reasonable number of interactions. The results also indicate that the algorithm is robust to minor inconsistencies in decision maker responses. There is also surprising robustness toward different patterns of interaction with the decision maker. The results are of interest to the evolutionary multiobjective (EMO) community actively developing hybrid interactive EMO approaches.