A Thurstonian view of the analytic hierarchy process

The application of deterministic decision models in situations characterized by noise and uncertainty is likely to produce results of questionable value. In this paper, some very simple probabilistic models are developed and substituted for the deterministic scales used in the Analytic Hierarchy Process (AHP). It is shown that the use of these probabilistic models can extend the domain of AHP to situations, such as consensual or group decision making, that possess significant amounts of uncertainty. In addition, explicit measures of the variation present in the evaluation of decision alternatives and attributes are obtained.     

Generalizing variance to allow the inclusion of decision attitude in decision making under uncertainty

The problem of decision making under uncertainty is considered. It is noted that an alternative is described in terms of an uncertainty profile. We observe that a major difficulty in the decision process is the comparison of these uncertainty profiles. We discuss the need for introducing some features of an uncertainty profile to help simplify this comparison. We note that the quantification of these simplifying features involves some subjective considerations about the decision makers preferences. We introduce the idea of the decision maker’s attitudinal character to help in the formulation of these considerations. We then investigate two important features associated with an uncertainty profile. The first, the representative value, is a generalization of expected value commonly used under probabilistic uncertainty. The second, called the measure of deviation, provides a generalization of the concept of variance. We show how these new measures allows us to consider uncertainty profiles other then just the probabilistic one. They also allow us introduce other decision maker attitudes then the one implicitly assumed with the expected value and variance.     

The evidential reasoning approach for multi-attribute decision analysis under interval uncertainty

The evidential reasoning (ER) approach is a method for multiple attribute decision analysis (MADA) under uncertainties. It improves the insightfulness and rationality of a decision making process by using a belief decision matrix (BDM) for problem modelling and the Dempster–Shafer (D–S) theory of evidence for attribute aggregation. The D–S theory provides scope and flexibility to deal with interval uncertainties or local ignorance in decision analysis, which is not explored in the original ER approach and will be investigated in this paper. Firstly, interval uncertainty will be defined and modelled in the ER framework. Then, an extended ER algorithm, IER, is derived, which enables the ER approach to deal with interval uncertainty in assessing alternatives on an attribute. It is proved that the original ER algorithm is a special case of the IER algorithm. The latter is demonstrated using numerical examples.     

DS/AHP method: A mathematical analysis,including an understanding of uncertainty

The analytic hierarchy process (AHP) was developed to aid decision makers to rank or sort information based on a number of criteria. A recent advance is the DS/AHP method which incorporates the Dempster–Shafer theory of evidence with AHP. This method allows judgements on groups of decision alternatives (DA) to be made, it also offers a measure of uncertainty in the final results. In this paper a mathematical analysis of DS/AHP is included, constructing the functional form of the preference weightings given to groups of DA. These functions allow an understanding of the appropriateness of the rating scale values used in the DS/AHP method, through evaluating the range of uncertainty able to be expressed by the decision maker.     

Modifications of the Hurwicz’s decision rule

The Hurwicz’s criterion is one of the classical decision rules applied in decision making under uncertainty as a tool enabling to find an optimal pure strategy both for interval and scenarios uncertainty. The interval uncertainty occurs when the decision maker knows the range of payoffs for each alternative and all values belonging to this interval are theoretically probable (the distribution of payoffs is continuous). The scenarios uncertainty takes place when the result of a decision depends on the state of nature that will finally occur and the number of possible states of nature is known and limited (the distribution of payoffs is discrete). In some specific cases the use of the Hurwicz’s criterion in the scenarios uncertainty may lead to quite illogical and unexpected results. Therefore, the author presents two new procedures combining the Hurwicz’s pessimism-optimism index with the Laplace’s approach and using an additional parameter allowing to set an appropriate width for the ranges of relatively good and bad payoffs related to a given decision. The author demonstrates both methods on the basis of an example concerning the choice of an investment project. The methods described may be used in each decision making process within which each alternative (decision, strategy) is characterized by only one criterion (or one synthetic measure).     

基于区间数期望和不确定性分析的多属性决策方法与应用

针对区间数多属性决策中的不确定性问题,提出基于区间数的不确定性分析方法,用集对分析联系数中的A表示区间数的数学期望,Bi表示区间数的不确定性,借助联系数A+Bi的加、乘运算建立区间数多属性决策模型,再利用i的不同取值进行不确定性分析.实例应用表明,方法算理清晰,算法简明,分析方便,结论可靠.     

Two axiomatic approaches to decision making using possibility theory

This paper proposes a utility theory for decision making under uncertainty that is described by possibility theory. We show that our approach is a natural generalization of the two axiomatic systems that correspond to pessimistic and optimistic decision criteria proposed by Dubois et al. The generalization is achieved by removing axioms that are supposed to reflect attitudes toward uncertainty, namely, pessimism and optimism. In their place we adopt an axiom that imposes an order on a class of canonical lotteries that realize either in the best or in the worst prize. We prove an expected utility theorem for the generalized axiomatic system based on the newly introduced concept of binary utility.     

Heterogeneous uncertainties in cholesterol management

Physicians use clinical guidelines to inform judgment about therapy. Clinical guidelines do not address three important uncertainties: (1) uncertain relevance of tested populations to the individual patient, (2) the patient’s uncertain preferences among possible outcomes, and (3) uncertain subjective and financial costs of intervention. Unreliable probabilistic information is available for some of these uncertainties; no probabilities are available for others. The uncertainties are in the values of parameters and in the shapes of functions. We explore the usefulness of info-gap decision theory in patient-physician decision making in managing cholesterol level using clinical guidelines. Info-gap models of uncertainty provide versatile tools for quantifying diverse uncertainties. Info-gap theory provides two decision functions for evaluating alternative therapies. The robustness function assesses the confidence—in light of uncertainties—in attaining acceptable outcomes. The opportuneness function assesses the potential for better-than-anticipated outcomes. Both functions assist in forming preferences among alternatives. Hypothetical case studies demonstrate that decisions using the guidelines and based on best estimates of the expected utility are sometimes, but not always, consistent with robustness and opportuneness analyses. The info-gap analysis provides guidance when judgment suggests that a deviation from the guidelines would be productive. Finally, analysis of uncertainty can help resolve ambiguous situations.     

A modified procedure for synthesising ratio judgements in the analytic hierarchy process

The analytic hierarchy process (AHP) has been widely applied to solve problems arising in group decision making, by synthesising different or conflicting judgements. However, directly synthesising conflicting judgements by calculating the geometric mean of preference weights (ratios) in AHP may not reach consensus from all members in a decision making group, especially, when those members represent the stakeholders of the decision making problem. This study proposes a new method that uses the genetic algorithm and utility function to synthesise preference weights to prevent this fallacy occurring when implementing the classical AHP approach. Using the proposed method, the final decision can be achieved with only minimally-adjusted preference weights.     

An analysis of distributions of priority values from alternative comparison scales within AHP

The analytic hierarchy process (AHP) introduced by T.L. Saaty is a well known and popular method of multi-criteria decision making. Central to this method are the pairwise comparisons between criteria (and decision alternatives) made using a 9-unit scale. The appropriateness of Saaty's original one-to-nine (1–9) scale has been the subject of much debate and cause for concern. This paper contrasts the appropriateness of the 1–9 scale with other alternative 9-unit scales also used in AHP, by looking at the probability distributions of the associated priority values. For large problems, estimated probability distributions are found for the priority values through using the method of Parzen Windows.     

Stochastic decision making using multiplicative AHP

The Analytic Hierarchy Process (AHP) has found a number of applications in decision making problems. Its multiplicative version, called the Multiplicative AHP (MAHP), has been proposed to overcome some of the criticisms of the conventional version. Both these methodologies operate by obtaining expert judgements on the ratios of perceived importance of objects under consideration. The literature on MAHP in dealing with these judgements, when they are specified without uncertainty, is well developed. However, stochastic aspects of these judgements have not received much consideration in the literature so far. Stochastic judgements are considered in this paper for use in MAHP. The fact that weight derivation in MAHP can be handled using mathematical programming is exploited and the literature on stochastic programming is adapted to the MAHP context.     

基于多维风险视角的政府卫生保健资源配置决策行为研究

研究在疾病风险和医学治疗风险同时共存的情形下,政府卫生保健资源的优先配置行为决策问题.当存在治疗风险(患病风险)时,分析病人患病的不确定性(治疗的不确定性)对政府卫生保健资源的配置效应,同时给出配置更多的卫生资源到更高风险病人群体的社会规划者的风险偏好条件.当这两类风险是局部的或正象限依赖的风险时,研究两种来源的风险对政府卫生保健资源的配置的联合影响.将之前学者提出的卫生保健资源的配置模型扩展到两类风险共存的情形,同时对于不确定下的卫生保健资源配置决策问题提供新的见解.     

指标排序条件下的多指标区间决策

提出了决策指标的权重不能完全确定而只能对权重进行大小排序,并且决策矩阵中的元素包含区间数的多指标决策问题.分4种类型给出区间数决策矩阵的规范化方法,给出了决策方案综合评价值区间的计算模型及算法:给出了区间数比较大小的可能度的概念及可能度的性质;给出了优序数的概念及有关定理.在此基础上,给出了一种简易且具有保序性的方案排序方法.最后应用实例对方法进行说明.     

Probabilistic judgments specified partially in the analytic hierarchy process

The Analytic Hierarchy Process (AHP) is a decision-making tool which yields priorities for decision alternatives. This paper proposes a new approach to elicit and synthesize expert assessments for the group decision process in the AHP. These new elicitations are given as partial probabilistic specifications of the entries of pairwise comparisons matrices. For a particular entry of the matrix, the partial probabilistic elicitations could arise in the form of either probability assignments regarding the chance of that entry falling in specified intervals or selected quantiles for that entry. A new class of models is introduced to provide methods for processing this partial probabilistic information. One advantage of this approach is that it allows to generate as many pairwise comparison matrices of the decision alternatives as one desires. This, in turn, allows us to determine the statistical significance of the priorities of decision alternatives.     

基于Vague集信息的多属性群决策专家水平评判方法

针对基于Vague集信息的多属性群决策专家水平评判问题提出了两种评判方法.首先引进了基于Vague集信息的多属性群决策信息体(即决策信息体)的相关概念,通过决策信息体构造了基于Vague集信息的一致性决策矩阵及模糊熵,其次利用Vague集信息的相似度量以及Vague集信息的模糊熵两种信息不确定性度量方法,对基于Vague集信息的多属性群决策专家水平评判问题提出了两种评判方法,即统计分析方法和模糊熵分析方法,对专家的评判水平进行排序.最后,通过一个算例说明两种方法的一致性、有效性和实用性.     

Using extended Monte Carlo simulation method for the improvement of risk management: Consideration of relationships between uncertainties

This paper considers the relationship of the major uncertainties of a project by using proposed approach. This approach by using rotary algorithm intellectualized the classic Monte Carlo simulation. This will help utility function to come closer to reality so that decision making and risk analysis would be done based on the real and possible modes, providing better conditions for decision making. Analyzing and investigating uncertainties are done in the risk management frame work. Because opportunities and threats are not separated, Monte Carlo simulation analysis is implemented as an integrated tool to reach the project goals, analyzing and investigating a variety of uncertainty permutations simultaneously. This method is a powerful tool for investigating the effects of all uncertainties’ occurrence, so it has noticeable benefits such as simultaneous consideration of uncertainties and the capability of representing several dimensions of utility function. In spite of these benefits, not considering the type and level of relationships, some permutations of uncertainties will occur that are not possible in real world. This would divert the utility function from reality. A simple example is used to illustrate the application of the model in practice.     

Fuzzy optimality based decision making under imperfect information without utility

In the realm of decision making under uncertainty, the general approach is the use of the utility theories. The main disadvantage of this approach is that it is based on an evaluation of a vector-valued alternative by means of a scalar-valued quantity. This transformation is counterintuitive and leads to loss of information. The latter is related to restrictive assumptions on preferences underlying utility models like independence, completeness, transitivity etc. Relaxation of these assumptions results into more adequate but less tractable models. In contrast, humans conduct direct comparison of alternatives as vectors of attributes’ values and don’t use artificial scalar values. Although vector-valued utility function-based methods exist, a fundamental axiomatic theory is absent and the problem of a direct comparison of vectors remains a challenge with a wide scope of research and applications. In the realm of multicriteria decision making there exist approaches like TOPSIS and AHP to various extent utilizing components-wise comparison of vectors. Basic principle of such comparison is the Pareto optimality which is based on a counterintuitive assumption that all alternatives within a Pareto optimal set are considered equally optimal. The above mentioned mandates necessity to develop new decision approaches based on direct comparison of vector-valued alternatives. In this paper we suggest a fuzzy Pareto optimality (FPO) based approach to decision making with fuzzy probabilities representing linguistic decision-relevant information. We use FPO concept to differentiate “more optimal” solutions from “less optimal” solutions. This is intuitive, especially when dealing with imperfect information. An example is solved to show the validity of the suggested ideas.     

An approach to the valuation and decision of ERP investment projects based on real options

The risks and uncertainties inherent in most enterprise resources planning (ERP) investment projects are vast. Decision making in multistage ERP projects investment is also complex, due mainly to the uncertainties involved and the various managerial and/or physical constraints to be enforced. This paper tackles the problem using a real-option analysis framework, and applies multistage stochastic integer programming in formulating an analytical model whose solution will yield optimum or near-optimum investment decisions for ERP projects. Traditionally, such decision problems were tackled using lattice simulation or finite difference methods to compute the value of simple real options. However, these approaches are incapable of dealing with the more complex compound real options, and their use is thus limited to simple real-option analysis. Multistage stochastic integer programming is particularly suitable for sequential decision making under uncertainty, and is used in this paper and to find near-optimal strategies for complex decision problems. Compared with the traditional approaches, multistage stochastic integer programming is a much more powerful tool in evaluating such compound real options. This paper describes the proposed real-option analysis model and uses an example case study to demonstrate the effectiveness of the proposed approach.     

Revising the OWA operator for multi criteria decision making problems under uncertainty

Multi criteria decision making (MCDM) problems are usually under uncertainty. One of these uncertain parameters is the decision maker (DM)’s degree of optimism, which has an important effect on the results. Fuzzy linguistic quantifiers are used to obtain the assessments of this parameter from DM and then, because of its uncertainty it is assumed to have stochastic nature. A new approach, entitled FSROWA, is introduced to combine the Fuzzy and Stochastic features into a Revised OWA operator.     

Using possibilistic logic for modeling qualitative decision: Answer Set Programming algorithms

A qualitative approach to decision making under uncertainty has been proposed in the setting of possibility theory, which is based on the assumption that levels of certainty and levels of priority (for expressing preferences) are commensurate. In this setting, pessimistic and optimistic decision criteria have been formally justified. This approach has been transposed into possibilistic logic in which the available knowledge is described by formulas which are more or less certainly true and the goals are described in a separate prioritized base. This paper adapts the possibilistic logic handling of qualitative decision making under uncertainty in the Answer Set Programming (ASP) setting. We show how weighted beliefs and prioritized preferences belonging to two separate knowledge bases can be handled in ASP by modeling qualitative decision making in terms of abductive logic programming where (uncertain) knowledge about the world and prioritized preferences are encoded as possibilistic definite logic programs and possibilistic literals respectively. We provide ASP-based and possibilistic ASP-based algorithms for calculating optimal decisions and utility values according to the possibilistic decision criteria. We describe a prototype implementing the algorithms proposed on top of different ASP solvers and we discuss the complexity of the different implementations.