RANK GENERATION,PRESERVATION, AND REVERSAL IN THE ANALYTIC HIERARCHY DECISION PROCESS

Decision making has the objective of finding the best alternative or set of alternatives by considering a number of goals, objectives, criteria, competitors, and other important factors. The analytic hierarchy process is a decision aid used to assist a decision maker in sorting out the complexity of a decision problem and making use of his or her judgments. A decision maker must be assured that the arithmetic operations of any such decision process are the right ones—that they surface the correct ranking and values of the alternatives and preserve or alter ranks appropriately when new alternatives are added or deleted. In this paper it will be shown that with absolute measurement, rank always is preserved, with relative measurement, rank changes with nspect to scveral criteria only because of the structural dependence (involving both numbers and measurements) of criteria on alternatives. A discussion of the effect on rank of replicas and near replicas of the alternatives also is given.     

DECISION MAKING INVOLVING SEQUENTIAL EVENTS: SOME EFFECTS OF DISAGGREGATED DATA AND DISPOSITIONS TOWARD RISK*

This paper concerns human processing of disaggregated data in a business decision setting involving sequential events. Subjects currently employed in trust banking institutions chose between two mutually exclusive alternatives in a series of decision trials. The outcome for each alternative was the result of a two-stage process; an unsuccessful outcome resulted from failure at either the first or second stage. The probabilities of success at each stage and the overall (joint) probability of success were provided for each alternative. Participants showed strongest preferences for an optimal (expected-value) strategy in processing the data; however, significant preferences for two suboptimal strategies also were observed. Preferences for suboptimal strategies were influenced by the magnitude of differences in the alternatives' joint probabilities of success, by the magnitude of differences in the alternatives' stage probabilities of success, and by subjects' dispositions toward risk.     

Modeling Correlated Discrete Uncertainties in Event Trees with Copulas

Modeling the dependence between uncertainties in decision and risk analyses is an important part of the problem structuring process. We focus on situations where correlated uncertainties are discrete, and extend the concept of the copula‐based approach for modeling correlated continuous uncertainties to the representation of correlated discrete uncertainties. This approach reduces the required number of probability assessments significantly compared to approaches requiring direct estimates of conditional probabilities. It also allows the use of multiple dependence measures, including product moment correlation, rank order correlation and tail dependence, and parametric families of copulas such as normal copulas, t‐copulas, and Archimedean copulas. This approach can be extended to model the dependence between discrete and continuous uncertainties in the same event tree.     

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.     

Physicians’ Labour Supply: The Wage Impact on Hours and Practice Combinations

Abstract. It is a stated aim to improve physician services in underserved sectors and areas. Increased wages is one instrument for boosting the hours provided by the personnel to the prioritized sub‐markets. This study applies an econometric framework that allows for non‐convex budget sets, non‐linear labour supply curves and imperfect markets with institutional constraints. The labour supply decision is viewed as a choice from a set of discrete alternatives (job packages) in a structural labour supply model estimated on Norwegian micro data. An out‐of‐sample prediction is also presented and evaluated by means of a natural experiment.     

PITFALLS OF ROUNDING IN DISCRETE MANAGEMENT DECISION PROBLEMS

A number of articles on managerial decision making have addressed the issue of whether or not to round a fractional solution to obtain a solution for a problem involving discrete alternatives. (An example is the problem in which the decision maker must select exactly one of several investment alternatives, but attaches no meaning to selecting two-thirds of one alternative and one-third of another.) Those articles which suggest that rounding can lead to undesirable answers are seemingly supported by the numerous “textbook examples” that purport to illustrate the dangers of rounding. However, the standard examples in which rounding fails to give a workable solution involve only a few rounding possibilities (usually two or four) and do not come from real world applications. Hence, it is questionable whether they provide any insight about what is likely to occur in a practical setting. This note fills a gap in previous discussions of rounding by providing two easily understood examples that dramatically portray the difficulties that rounding can encounter. The first example belongs to an important class of practical problems. We illustrate that rounding fails not only for this example, but also fails for all problems in its class. The second example is a unique “showcase” problem which can be summarized by a 5 times 5 cost matrix. This problem contains more than a million rounding alternatives, all of them infeasible! Following these examples, we present a “rounding paradox” and we show that its resolution gives analytical support to the conclusion that rounding will produce grave difficulties in a wide variety of practical situations.     

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。     

区间概率条件下属性值为不确定语言变量的风险型多属性决策研究

针对区间概率条件下属性值为不确定语言信息且属性权重未知的风险型多属性决策问题,提出了一种基于概率理论和不确定语言变量的TOPSIS决策方法。首先建立了区间概率转化为点概率的数学模型,通过期望值将风险型决策矩阵转化为确定型矩阵;然后利用方案与理想解越近方案越优,与负理想解越远方案越优的原则建立属性权重确定模型,并利用TOPSIS方法的相对优属度大小确定方案的排序;最后通过应用案例说明了本方法的决策步骤。     

INFLUENCE OF MISSING ATTRIBUTES ON SELECTING A BEST MULTIATTRIBUTED ALTERNATIVE

Alternatives involving many factors arc difficult to evaluate because of multiple underlying competing objectives. If evaluation is based on an incomplete set of factors, and if the purpose of the evaluation is to select a single overall best alternative, inferior alternatives may be selected with surprising frequency and/or severe negative impact. At the same time, sensitivity analysis of evaluation scores based on statistical criteria can easily mask the impact and the frequency of selection of inferior alternatives. In this paper, criteria appropriate to reflect the decision impact are developed and both the frequency and impact of the selection of inferior alternatives are demonstrated empirically. Previous studies based on statistical criteria [1] [9] indicated minimal impact on overall evaluation and selection. This paper demonstrates that high statistical criterion values coexist with frequent and/or serious errors of selection.     

Estimating Moments of Subjectively Assessed Distributions*

Moment‐matching discrete distributions were developed by Miller and Rice (1983) as a method to translate continuous probability distributions into discrete distributions for use in decision and risk analysis. Using gaussian quadrature, they showed that an n‐point discrete distribution can be constructed that exactly matches the first 2n ‐ 1 moments of the underlying distribution. These moment‐matching discrete distributions offer several theoretical advantages over the typical discrete approximations as shown in Smith (1993), but they also pose practical problems. In particular, how does the analyst estimate the moments given only the subjective assessments of the continuous probability distribution? Smith suggests that the moments can be estimated by fitting a distribution to the assessments. This research note shows that the quality of the moment estimates cannot be judged solely by how close the fitted distribution is to the true distribution. Examples are used to show that the relative errors in higher order moment estimates can be greater than 100%, even though the cumulative distribution function is estimated within a Kolmogorov‐Smirnov distance less than 1%.     

Optimal Selection Strategy for the Group Interview Problem

We propose a more generalized version of the secretary problem, called the group interview problem, in which each group contains several alternatives and each group of alternatives is presented and evaluated sequentially over time. Using the assumptions corresponding to the classical secretary problem, we derive an optimal selection strategy which maximizes the probability of winning or selecting the single best choice in a given sequence of groups. We further address the problem of choosing at the beginning of the evaluation process a sequence of groups to maximize the winning probability. Because of formidable computational requirements to obtain an optimal solution to this sequencing problem, we then develop a heuristic algorithm based on several properties inherent in an optimal selection strategy. The heuristic procedure is evaluated experimentally using Monte Carlo simulation and is shown to be effective in obtaining near-optimal (within 5 percent) solutions.     

A Note on “Discrete Sequential Search with Group Activities”

This note comments on a paper published by Wagner and Davis [Decision Sciences (2001), 32(4), 557–573]. These authors present an integer‐programming model for the single‐item discrete sequential search problem with group activities. Based on their experiments, they conjecture that the problem can be solved as a linear program. In this note, we provide a counterexample for which the optimal value of the linear program they propose is different from the optimal value of the integer‐programming model, hence contradicting their conjecture for the specific linear program that they specify. To the best of our knowledge, the conjecture settled in this note was still an open question.     

Processing in a multi-machine batch manufacturing system with sequence dependent cost

Many workcells in batch manufacturing systems are populated with multiple, nonidentical machines that perform similar tasks. Because of the size of a batch when a job arrives, it may be uneconomical to set up two or more machines to process the same job simultaneously. An economic decision has to be made as regards which machine in the cell to assign the job. Likewise, many multi-operation jobs can be processed using one of several feasible operation sequences that may lead to different total manufacturing costs. The cost differences are the result of several factors, among which are processing time and cost dependencies between operations, fixturing requirements, and material handling requirements. When the workcell machine selection decision is considered along with the operation sequencing decision, determination of the best machine in a cell and the best operation sequence for the batch is a non-trivial task. In this paper, we address the problem of selecting the best machine within a cell and the best operation sequence for a batch when operation cost is machine and sequence dependent. The problem is modeled mathematically and solved using a heuristic algorithm. The performance of the algorithm is compared with that of an exact solution procedure.     

On the Sequential Selection Problem*

Kwan and Yuan [13] considered the sequential selection problem in which an employer should arrange the sequence of interviews with job applicants to fill a position. In this note, it is shown that their selection problem would be alternately interpreted as an optimal search problem or, more specifically, a discrete search problem with a stationary target. A more generalized version of the ordering problem is proposed which explicitly considers the time value of money. Also, the optimality of the ordering strategy in the generalized problem is proven by the pair-wise exchange method, which is simpler than the induction hypothesis-based proof. The sequention selection problem is shown to be a special case of the general ordering problem where the discount rate is zero.     

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 WTP‐Choice Model: Empirical Validation,Competitive and Centralized Pricing

Willingness To Pay (WTP) of customers plays an anchoring role in pricing. This study proposes a new choice model based on WTP, incorporating sequential decision making, where the products with positive utility of purchase are considered in the order of customer preference. We compare WTP‐choice model with the commonly used (multinomial) Logit model with respect to the underlying choice process, information requirement, and independence of irrelevant alternatives. Using WTP‐choice model, we find and compare equilibrium and centrally optimal prices and profits without considering inventory availability. In addition, we compare equilibrium prices and profits in two contexts: without considering inventory availability and under lost sales. One of the interesting results with WTP‐choice model is the “loose coupling” of retailers in competition; prices are not coupled but profits are. That is, each retailer should charge the monopoly price as the collection of these prices constitute an equilibrium but each retailer's profit depends on other retailers' prices. Loose coupling fails with dependence of WTPs or dependence of preference on prices. Also, we show that competition among retailers facing dependent WTPs can cause price cycles under some conditions. We consider real‐life data on sales of yogurt, ketchup, candy melt, and tuna, and check if a version of WTP‐choice model (with uniform, triangle, or shifted exponential WTP distribution), standard or mixed Logit model fits better and predicts the sales better. These empirical tests establish that WTP‐choice model compares well and should be considered as a legitimate alternative to Logit models for studying pricing for products with low price and high frequency of purchase.     

Selecting sourcing partners for a make-to-order supply chain

Make-to-order manufacturers face an idiosyncratic and complex situation in the sourcing selection process. Although they can source from several key suppliers, each order requires custom key parts that cannot be stocked. Our research provides a decision model to facilitate the sourcing process for these manufacturers using information about their sourcing partners’ cost and processing time. The manufacturer can calculate the total cost and on-time probability for all possible combinations of certified suppliers and key part allotments to obtain a sourcing portfolio with several sourcing alternatives for a desired service level. The portfolio allows the manufacturer to make trade-offs between cost and reliability to finish the job on time. Additionally, the portfolio can be obtained for a given due date or for reduced due dates in a competitive bid situation. The portfolio approach allows the manufacturer to maintain control over the sourcing selection process by partnering with sourcing members to keep costs low without losing the needed flexibility to meet customer demands.     

考虑决策者损失规避的异质信息多属性变权决策方法

现实中存在大量异质信息（或数据）和需要考虑权重随属性值变化的多属性决策问题。针对这类异质信息多属性决策问题，本文提出了一种基于前景理论的变权综合求解方法。首先，构建了异质信息的统一距离计算公式，进而计算各个决策方案的相对贴近度；然后，提出基于不同类型效用函数的变权向量构造方法；其次，以初始权重为参考点，计算变权向量相对于参考点的益损决策矩阵，进而计算考虑决策者权重损失和收益的风险态度的各个决策方案的前景综合值，据此确定方案优劣排序和最优方案。通过数值例子的计算分析说明，文中所提决策模型与方法具有较好的有效性和合理性，可为解决复杂情景的决策问题提供理论依据与方法支持。     

Managing Potential Health Risks from Electric Powerlines: A Decision Analysis Caught in Controversy

Over the past 20 years, several epidemiological studies have found an association between exposure to electromagnetic fields (EMFs) and health effects, including childhood leukemia and adult brain cancer. However, experts strongly disagree about whether this association is causal and, if so, how strong it is. In this article, we examine several alternatives to reduce EMFs from sources of the California power grid, including undergrounding distribution and transmission lines and reconfiguring or rephasing lines. The alternatives were evaluated in terms of the potential health risk reduction, cost, impacts on service reliability, property values, and many other consequences. Because of the uncertainty about an EMF-health link, the main effort was to determine the sensitivity of the decisions to the probability and seriousness of an EMF hazard. User-friendly computer models were developed to allow stakeholders to change the model assumptions and parameters to analyze the impacts of their own assumptions and estimates on the decision. The analysis clearly demonstrated that only four of the many concerns raised by the stakeholders could make a difference in the decision: health risks, costs, service reliability, and property values. Whether undergrounding, moderate alternatives for EMF reduction, or no change was the best decision depended on a few key factors, including the probability that EMF exposure is a hazard, the severity of this hazard, how the EMF reduction measures are financed, and the impacts on property values. While the analysis did not resolve the EMF issues, it showed that even in the most controversial settings, a little analysis goes a long way to clarifying the issues and to focus the debate.