ELICITATION OF SUBJECTIVE PROBABILITIES IN THE CONTEXT OF DECISION-MAKING *

In this paper a framework is developed to seek information from the decision-maker about uncertain events, and to use this information for identifying a preferred alternative. It is shown how a preferred alternative can be identified with less than perfect knowledge of the probabilities.     

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.     

Entropy Principles in Decision Making Under Risk

This paper considers decision problems where: (1) The exact probability distribution over the states of nature is not precisely known, but certain prior information is available about the possibilities of these outcomes; (2) A prior distribution over the states of nature is known, but new constraint information about the probabilities becomes available. The maximum entropy principle asserts that the probability distribution with maximum entropy, satisfying the prior knowledge, should be used in the decision problem. The minimum cross-entropy principle says that the posterior distribution is the one which minimizes cross-entropy, subject to the new constraint information. The entropy principles have not gone uncriticized, and this literature, together with that justifying the principles, is surveyed. Both principles are illustrated in a number of situations where the distribution is either discrete or continuous. The discrete distribution case with prior interval estimates based on expert opinions is considered in detail.     

The analytic hierarchy process with interval preference statements

In the analytic hierarchy process (AHP), interval judgments instead of precise ratios are widely accepted and can be practically used to solve decision-making problems when uncertainty exists because of scant information available or insufficient understanding of the problem. This paper presents a simple and effective method for finding the extreme points in a range of interval ratios (such as loose articulation, minimum number of interval ratios, and general interval ratios) and ultimately for establishing the dominance relations among alternatives using the identified extreme points. This is followed by an enumeration or simulation approach to manage situations in which the best or a full ranking of alternatives remains unidentified.     

Configuration of flexibility strategies under supply uncertainty

In this paper, we study a buyer׳s configuration of flexibility strategies under supply uncertainty. His main supplier׳s production process is uncertain, and he can either choose pricing flexibility (setting prices depending on the available supply) or operational flexibility (requesting a contingent order from a backup supplier). As the buyer may or may not find a suitable contingent supplier ex post, we study two scenarios that the backup supplier׳s supply is infinite, and that this supply is random. We also include the factor that the main supplier may determine the wholesale price. We demonstrate that the adoption of flexibility strategies is controlled by threshold policies in different scenarios whether the main supplier determines the wholesale price or not. We also investigate how the buyer׳s attribute (finding a suitable contingent supplier) affects the configuration of flexibility strategies.     

Enhancing corporate social responsibility: Contract design under information asymmetry

We consider a two-stage supply chain in which a contract manufacturer (CM) sells products through a brand name retailer. The contract manufacturer can invest in corporate social responsibility (CSR) activities to improve customer perception about the firm and increase demand, while the retailer can influence the demand by exerting marketing efforts. We design optimal contracts for such a supply chain, which faces information asymmetry. The wholesale price contract was developed as the base model to derive insight into the value of information sharing. We examine the impact of CSR cost on CSR commitment and profits. We find that CM׳s CSR cost impacts the CM׳s and the retailer׳s profits differently. Under certain conditions, the CM׳s profit will increase with cost, while that of the retailer is uncertain. We also propose two-part tariff contracts for both the symmetric and asymmetric cases with the aim of maximizing the retailer׳s profit and improving CM׳s commitment to CSR. Finally, numerical experiments are conducted to illustrate and validate the proposed models and provide managerial insights.     

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.     

Probability Elicitation Under Severe Time Pressure: A Rank‐Based Method

Probability elicitation protocols are used to assess and incorporate subjective probabilities in risk and decision analysis. While most of these protocols use methods that have focused on the precision of the elicited probabilities, the speed of the elicitation process has often been neglected. However, speed is also important, particularly when experts need to examine a large number of events on a recurrent basis. Furthermore, most existing elicitation methods are numerical in nature, but there are various reasons why an expert would refuse to give such precise ratio‐scale estimates, even if highly numerate. This may occur, for instance, when there is lack of sufficient hard evidence, when assessing very uncertain events (such as emergent threats), or when dealing with politicized topics (such as terrorism or disease outbreaks). In this article, we adopt an ordinal ranking approach from multicriteria decision analysis to provide a fast and nonnumerical probability elicitation process. Probabilities are subsequently approximated from the ranking by an algorithm based on the principle of maximum entropy, a rule compatible with the ordinal information provided by the expert. The method can elicit probabilities for a wide range of different event types, including new ways of eliciting probabilities for stochastically independent events and low‐probability events. We use a Monte Carlo simulation to test the accuracy of the approximated probabilities and try the method in practice, applying it to a real‐world risk analysis recently conducted for DEFRA (the U.K. Department for the Environment, Farming and Rural Affairs): the prioritization of animal health threats.     

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.     

A MATHEMATICAL PROGRAMMING APPROACH TO SENSITIVITY ANALYSIS IN SINGLE-STAGE DECISION MAKING

This paper presents sensitivity analyses for a single-stage decision problem where an action which will maximize the expected payoff must be chosen from a finite number of actions given the states of nature, their probabilities, and the payoffs corresponding to each action and state of nature. Three types of sensitivity analysis are developed: (1) sensitivity analysis on the probabilities keeping the payoff numbers fixed, (2) sensitivity analysis on the payoffs keeping the probabilities fixed, and (3) joint sensitivity analysis on the payoffs and the probabilities. The approach is illustrated with an example. Quite often the sensitivity analysis can be conducted by solving an appropriate linear or quadratic programming problem.     

Decision theory,linear partial information and statistical dominance

The paper deals with decision making under conditions of linear partial information, i.e. when probabilities of states of nature are not known precisely, but are subject to linear constraints. Conditions ensuring strict and weak statistical dominance of one strategy over another are derived. It is also shown that weak dominance in terms of payoffs is equivalent to weak dominance in terms of regrets. The new results are more general than those obtained by Fishburn and by Kmietowicz and Pearman for weak and strict ranking of probabilities, and include them as special cases. The new results can be employed in practical decision making in several ways.     

Supplier competition and its impact on firm׳s sourcing strategy

Most research on firms׳ sourcing strategies assumes that wholesale prices and reliability of suppliers are exogenous. It is of our interest to study suppliers׳ competition on both wholesale price and reliability and firms׳ corresponding optimal sourcing strategy under complete information. In particular, we study a problem in which a firm procures a single product from two suppliers, taking into account suppliers׳ price and reliability differences. This motivates the suppliers to compete on these two factors. We investigate the equilibria of this supplier game and the firm׳s corresponding sourcing decisions. Our study shows that suppliers׳ reliability often plays a more important role than wholesale price in supplier competition and that maintaining high reliability and a high wholesale price is the ideal strategy for suppliers if multiple options exist. The conventional wisdom implies that low supply reliability and high demand uncertainty motivate dual-sourcing. We notice that when the suppliers׳ shared market/transportation network is often disrupted and demand uncertainty is high, suppliers׳ competition on both price and reliability may render the sole-sourcing strategy to be optimal in some cases that depend on the format of suppliers׳ cost functions. Moreover, numerical study shows that when the cost or vulnerability (to market disruptions) of one supplier increases, its profit and that of the firm may not necessarily decrease under supplier competition.     

Information sharing in a supply chain with a make-to-stock manufacturer

We study ex ante information sharing in a supply chain consisting of a downstream retailer and a make-to-stock upstream manufacturer. The retailer has imperfect demand information and may choose to share it with the manufacturer. Based on the information sharing arrangement, the manufacturer makes the wholesale price and the stocking level decisions. Then the retailer decides the order quantity and the manufacturer fulfills the order up to the available stock level. We find that the retailer has an incentive to voluntarily share the information with the make-to-stock manufacturer if the magnitude of demand uncertainty is intermediate. This stands in sharp contrast with the existing studies which show that the retailer never shares information when the manufacturer is make-to-order. Our results highlight the interdependence between the retailer׳s incentive to share information and the manufacturer׳s operational and marketing decisions.     

Integrated framework for robustness analysis using ratio-based efficiency model with application to evaluation of Polish airports

We consider a problem of evaluating efficiency of Decision Making Units (DMUs) based on their deterministic performance on multiple consumed inputs and multiple produced outputs. We apply a ratio-based efficiency measure, and account for the Decision Maker׳s preference information representable with linear constraints involving input/output weights. We analyze the set of all feasible weights to answer various robustness concerns by deriving: (1) extreme efficiency scores and (2) extreme efficiency ranks for each DMU, (3) possible and necessary efficiency preference relations for pairs of DMUs, (4) efficiency distribution, (5) efficiency rank acceptability indices, and (6) pairwise efficiency outranking indices. The proposed hybrid approach combines and extends previous results from Ratio-based Efficiency Analysis and the SMAA-D method. The practical managerial implications are derived from the complementary character of accounted perspectives on DMUs׳ efficiencies. We present an innovative open-source software implementing an integrated framework for robustness analysis using a ratio-based efficiency model on the diviz platform. The proposed approach is applied to a real-world problem of evaluating efficiency of Polish airports. We consider four inputs related to the capacities of a terminal, runways, and an apron, and to the airport׳s catchment area, and two outputs concerning passenger traffic and number of aircraft movements. We present how the results can be affected by integrating the weight constraints and eliminating outlier DMUs.     

Concerns,Challenges, and Directions of Development for the Issue of Representing Uncertainty in Risk Assessment

In the analysis of the risk associated to rare events that may lead to catastrophic consequences with large uncertainty, it is questionable that the knowledge and information available for the analysis can be reflected properly by probabilities. Approaches other than purely probabilistic have been suggested, for example, using interval probabilities, possibilistic measures, or qualitative methods. In this article, we look into the problem and identify a number of issues that are foundational for its treatment. The foundational issues addressed reflect on the position that “probability is perfect” and take into open consideration the need for an extended framework for risk assessment that reflects the separation that practically exists between analyst and decisionmaker.     

SENSITIVITY ANALYSIS IN DECISION THEORY

This paper explores linear programming-like sensitivity analysis in decision theory. In particular, the paper considers the sensitivity of an optimal decision to changes in probabilities of the states of nature and the development of “confidence spheres” to bound arbitrary parametric changes in the probability vector. Such information can be used to assess the accuracy required in assigning probabilities and the confidence in the maximumutility decision.     

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).     

动态联盟中伙伴选择问题的区间规划模型及其求解

伙伴选择和风险管理是动态联盟中的重要决策问题,当考虑失败风险时,失败概率无法给出精确值,因此,考虑采用不确定性规划描述此类问题.提出动态联盟中伙伴选择问题的区间规划模型,模型中用区间数表示联盟伙伴的失败概率.为了求解该模型,引入序关系,并利用Nakahara和Ishibuchi的定理,将区间规划模型转化为等价的清晰双目标模型.设计带自适应适值函数的遗传算法,求出问题的全部非劣解.经过对多个问题的仿真,证明了算法的有效性.     

需求分布规律变化情况下的报童订货策略

在需求分布规律变化情况下,报童在进行订货决策时会因为错误判断需求分布规律而导致期望库存成本增加。为了解决这一问题,本文集成传统历史需求信息和非传统需求信息以正确地认知需求分布规律,在此基础上决策订货量。假设需求服从均值不同、方差相同的两种类型的正态分布,每一种正态分布的概率已知。利用信号检测理论构建基于历史需求信息与需求分布概率的报童最优订货策略,并与只基于需求分布概率的直觉规则订货策略进行对比。结果表明：只要排除需求分布概率很大或很小两种极端情况,最优订货策略比直觉规则订货策略在控制期望库存成本方面的作用更明显,即利用历史需求信息可以有效修正报童对实际需求分布的检测结果,从而提高实际订货决策的准确性。研究结果对传统历史需求信息和非传统需求信息的集成以及需求信息交换等有一定的管理学启示和应用价值。