Implementing stochastic multicriteria acceptability analysis

Stochastic multicriteria acceptability analysis (SMAA) is a family of methods for aiding multicriteria group decision making in problems with inaccurate, uncertain, or missing information. These methods are based on exploring the weight space in order to describe the preferences that make each alternative the most preferred one, or that would give a certain rank for a specific alternative. The main results of the analysis are rank acceptability indices, central weight vectors and confidence factors for different alternatives. The rank acceptability indices describe the variety of different preferences resulting in a certain rank for an alternative, the central weight vectors represent the typical preferences favouring each alternative, and the confidence factors measure whether the criteria measurements are sufficiently accurate for making an informed decision.     

On rank reversal in decision analysis

Analytic hierarchy process (AHP) has been criticized for its possible rank reversal phenomenon caused by the addition or deletion of an alternative. This paper shows the fact that the rank reversal phenomenon occurs not only in the AHP but also in many other decision making approaches such as the Borda–Kendall (BK) method for aggregating ordinal preferences, the simple additive weighting (SAW) method, the technique for order preference by similarity to ideal solution (TOPSIS) method, and the cross-efficiency evaluation method in data envelopment analysis (DEA). Numerical examples are provided to illustrate the rank reversal phenomenon in these popular decision making approaches.     

一种PROMETHEE Ⅱ权重的敏感性分析方法

以往MADM的敏感性分析主要研究的是使方案集排序稳定的参数区间。本文针对PROMETHEEⅡ方法的权重建立一种新的敏感性分析数学模型，利用经典的线性规划方法，求解使得某方案排序第一且变化最小的权重值，回答了权重超出稳定区间后排序改变方向的问题。在实际应用中，有利于帮助决策者及时调整权重，得到合理结果。     

基于方案贴近度和满意度的交互式不确定多属性决策方法

针对属性权重信息部分确知且对方案有偏好的不确定多属性决策问题,提出一种基于方案贴近度和满意度的交互式决策方法.方法首先利用已知的客观信息和决策者的主观要求建立单目标规划模型,其次通过对方案满意度和综合度的给定与修正来实现人机交互决策.最后,通过实例说明模型及方法的可行性和有效性.     

MOIRA: A decision support system for decision making on aquatic ecosystems contaminated by radioactive fallout

Interventions to restore radionuclide contaminated aquatic ecosystems may reduce individual and collective radiation doses, but may also result in detrimental ecological, social and economic effects. Decision makers must carefully evaluate possible impacts before choosing a countermeasure, hence decision analysis methods constitute an important aid to rank intervention strategies after the contamination of an aquatic ecosystem. We describe MOIRA, a decision support system for the identification of optimal remedial strategies to restore water systems after accidental introduction of radioactive substances. MOIRA includes an evaluation module based on a multi-attribute value model to rank alternatives and a module to perform multiparametric sensitivity analyses, both with respect to weights and values, to allow us to gain insights into the problem. The problem is under certainty since the validation of models used to quantify countermeasure impacts suggests little uncertainty in policy effects.The system is implemented in a PC based decision support system which allows the inclusion of all relevant information.     

An improved method for ranking alternatives in multiple criteria decision analysis

Scoring rules are an important disputable subject in data envelopment analysis (DEA). Various organizations use voting systems whose main object is to rank alternatives. In these methods, the ranks of alternatives are obtained by their associated weights. The method for determining the ranks of alternatives by their weights is an important issue. This problem has been the subject at hand of some authors. We suggest a three-stage method for the ranking of alternatives. In the first stage, the rank position of each alternative is computed based on the best and worst weights in the optimistic and pessimistic cases, respectively. The vector of weights obtained in the first stage is not a singleton. Hence, to deal with this problem, a secondary goal is used in the second stage. In the third stage of our method, the ranks of the alternatives approach the optimistic or pessimistic case. It is mentionable that the model proposed in the third stage is a multi-criteria decision making (MCDM) model and there are several methods for solving it; we use the weighted sum method in this paper. The model is solved by mixed integer programming. Also, we obtain an interval for the rank of each alternative. We present two models on the basis of the average of ranks in the optimistic and pessimistic cases. The aim of these models is to compute the rank by common weights.     

部分权重信息下对方案有互补偏好的多属性决策法

研究了只有部分权重信息且对方案的偏好信息以模糊互补判断矩阵形式给出的多属性决策问题.首先,基于模糊互补判断矩阵的主观偏好信息,利用转换函数将客观决策信息一致化,建立一个目标规划模型,通过求解该模型得到属性权重,从而利用加性加权法获得各方案的综合属性值,并以此对方案进行排序或择优.提出了一种基于目标规划的多属性决策方法.该方法具有操作简便和易于上机实现的特点.最后,通过实例说明模型及方法的可行性和有效性.     

A sensitivity analysis algorithm for hierarchical decision models

In this paper, a comprehensive algorithm is developed to analyze the sensitivity of hierarchical decision models (HDM), including the analytic hierarchy process and its variants, to single and multiple changes in the local contribution matrices at any level of the decision hierarchy. The algorithm is applicable to all HDM that use an additive function to derive the overall contribution vector. It is independent of pairwise comparison scales, judgment quantification techniques and group opinion combining methods. The allowable range/region of perturbations, contribution tolerance, operating point sensitivity coefficient, total sensitivity coefficient and the most critical decision element at a certain level are identified in the HDM SA algorithm. An example is given to demonstrate the application of the algorithm and show that HDM SA can reveal information more significant and useful than simply knowing the rank order of the decision alternatives.     

Multi-attribute group decision-making with multi-granularity linguistic assessment information: An improved approach based on deviation and TOPSIS

With respect to group decision-making problems with multi-granularity linguistic assessment information, a new approach is proposed. Firstly, the computational formulae are given in order to transform and unify the multi-granularity linguistic comparison matrices. Secondly, the method of standard and mean deviation is applied to determine the unknown attribute weights, and the weights of the decision makers will be determined by using the extended TOPSIS (technique for order preference by similarity to an ideal solution) method. Finally, based on the LWAA (linguistic weighted arithmetic averaging) operator, information on the preference provided by each decision maker is aggregated into the comprehensive evaluation value of each alternative, and the most desirable alternative is selected. The proposed approach expands the research in multi-attribute group decision-making with multi-granularity linguistic assessment information by both considering the weights of the attributes and decision makers, and objective weighting for them. A numerical example is given to illustrate the practicability and usefulness of the proposed approach.     

Simulation techniques for the sensitivity analysis of multi-criteria decision models

This paper presents a simulation approach for high dimensional sensitivity analysis of the weights of multi-criteria decision models. This approach allows simultaneous changes of the weights and generates results that can easily be analyzed statistically to provide insights into multi-criteria model recommendations. In this study we consider three cases: no information, order information, and partial information regarding the weights. Our approach also allows investigation of sensitivity to the form of multi-criteria decision models. The simulation procedures we propose can also be used to aide in the actual decision process, particularly when the task is to select a subset of superior alternatives.     

An extended maximin approach for decision analysis with uncontrollable factors

In this paper an extension to the maximin approach to decision analysis in the presence of uncontrollable factors is proposed. This extension is based on the assumption that probabilities of consequences are known. Using the language of stochastic dominance, one decision alternative is preferred to another if the cumulative distribution function of the first alternative dominates that of the second in some area of low value consequences. This approach is an extension of a standard lexicographic maximin procedure to a case in which decision alternatives are characterised by arbitrary, including continuous, sets of consequences. Applications of the suggested approach to an ‘attack–defence’ type game and to the problems of location of public facilities are discussed.     

Postoptimal analysis in a multi-attribute decision model for restoring contaminated aquatic ecosystems

We describe the evaluation module of the MOIRA system, developed to identify optimal remedial strategies for restoring radionuclide contaminated aquatic ecosystems and drainage areas. This module includes a multiparametric sensitivity analysis, which is based on a multi-attribute additive value model, aimed at identifying optimal remedial strategies for restoring aquatic ecosystems contaminated by radionuclides. We introduce the sensitivity analysis to check the robustness of the conclusions on the inputs. This provides insights into the problem in the sense of making better use of the available information. This analysis is focused on judgemental inputs, imprecise value functions on attributes and imprecise scaling factors or weights for their aggregation. These are of utmost importance in determining the optimal countermeasures.     

Pseudo-criteria versus linear utility function in stochastic multi-criteria acceptability analysis

Stochastic multi-criteria acceptability analysis (SMAA) is a multi-criteria decision support method for multiple decision-makers (DMs) in discrete problems. SMAA does not require explicit or implicit preference information from the DMs. Instead, the method is based on exploring the weight space in order to describe the valuations that would make each alternative the preferred one. Partial preference information can be represented in the weight space analysis through weight distributions. In this paper we compare two variants of the SMAA method using randomly generated test problems with 2–12 criteria and 4–12 alternatives. In the original SMAA, a utility or value function models the DMs' preference structure, and the inaccuracy or uncertainty of the criteria is represented by probability distributions. In SMAA-3, ELECTRE III-type pseudo-criteria are used instead. Both methods compute for each alternative an acceptability index measuring the variety of different valuations that supports this alternative, and a central weight vector representing the typical valuations resulting in this decision. We seek answers to three questions: (1) how similar are the results provided by the decision models, (2) what kind of systematic differences exists between the models, and (3) how could one select indifference and preference thresholds of the pseudo-criteria model to match a utility model with given probability distributions?     

Decision analysis with indeterminate or incoherent probabilities

This paper presents a new method of modeling indeterminate and incoherent probability judgments in decision analysis problems. The decision maker's degree of beliefs in the occurrence of an event is represented by a unimodal (in fact, concave) function on the unit interval, whose parameters are elicited in terms of lower and upper probabilities with attached confidence weights. This is shown to provide a unified framework for performing sensitivity analysis, reconciling incoherence, and combining expert judgments.This material is based upon work supported by the National Science Foundation under grant SES-85-11675, and by the Business Associates Fund at the Fuqua School of Business.     

基于累积前景理论和VIKOR的毕达哥拉斯犹豫模糊风险型多属性决策方法

针对属性权重未知,且属性值为毕达哥拉斯犹豫模糊数(PHFN)的风险型多属性决策问题,考虑到决策者的有限理性行为,提出基于累积前景理论(CPT)和多准则妥协优化解(VIKOR)的决策方法。首先,定义PHFN的分散率,并构建优化模型确定属性权重。其次,将CPT融入PHFN环境,定义PHFN的价值函数,并结合决策权重函数计算方案在各属性下的综合前景值。进一步,构建综合前景值矩阵,在此基础上运用VIKOR法确定方案排序。最后,通过风险投资项目选择的应用案例说明所提方法是可行、有效的。     

Robust optimization analysis for multiple attribute decision making problems with imprecise information

This paper considers ranking decision alternatives under multiple attributes with imprecise information on both attribute weights and alternative ratings. It is demonstrated that regret results from the decision maker??s inadequate knowledge about the true scenario to occur. Potential optimality analysis is a traditional method to evaluate alternatives with imprecise information. The essence of this approach is to identify any alternative that outperforms the others in its best-case scenario. Our analysis shows that potential optimality analysis is optimistic in nature and may lead to a significant loss if an unfavorable scenario occurs. We suggest a robust optimization analysis approach that ranks alternatives in terms of their worst-case absolute or relative regret. A robust optimal alternative performs reasonably well in all scenarios and is shown to be desirable for a risk-concerned decision maker. Linear programming models are developed to check robust optimality.     

Rank reversals in multi-criteria decision analysis with statistical modelling of ratio-scale pairwise comparisons

In multi-criteria decision analysis, the overall performance of decision alternatives is evaluated with respect to several, generally conflicting decision criteria. One approach to perform the multi-criteria decision analysis is to use ratio-scale pairwise comparisons concerning the performance of decision alternatives and the importance of decision criteria. In this approach, a classical problem has been the phenomenon of rank reversals. In particular, when a new decision alternative is added to a decision problem, and while the assessments concerning the original decision alternatives remain unchanged, the new alternative may cause rank reversals between the utility estimates of the original decision alternatives. This paper studies the connections between rank reversals and the potential inconsistency of the utility assessments in the case of ratio-scale pairwise comparisons data. The analysis was carried out by recently developed statistical modelling techniques so that the inconsistency of the assessments was measured according to statistical estimation theory. Several type of decision problems were analysed and the results showed that rank reversals caused by inconsistency are natural and acceptable. On the other hand, rank reversals caused by the traditional arithmetic-mean aggregation rule are not in line with the ratio-scale measurement of utilities, whereas geometric-mean aggregation does not cause undesired rank reversals.     

A new approach for weight derivation using data envelopment analysis in the analytic hierarchy process

Recently, some researches have been carried out in the context of using data envelopment analysis (DEA) models to generate local weights of alternatives from pairwise comparison matrices used in the analytic hierarchy process (AHP). One of these models is the DEAHP. The main drawback of the DEAHP is that it generates counter-intuitive priority vectors for inconsistent pairwise comparison matrices. To overcome the drawbacks of the DEAHP, this paper proposes a new procedure entitled Revised DEAHP, and it will be shown that this procedure generates logical weights that are consistent with the decision maker's judgements and is sensitive to changes in data of the pairwise comparison matrices. Through a numerical example, it will be shown that the Revised DEAHP not only produces correct weights for inconsistent matrices but also does not suffer from rank reversal when an irrelevant alternative is added or removed.     

Dominance and potential optimality in multiple criteria decision analysis with imprecise information

This paper discusses multiple criteria models of decision analysis with finite sets of alternatives. A weighted sum of criteria is used to evaluate the performance of alternatives. Information about the weights is assumed to be in the form of arbitrary linear constraints. Conditions for checking dominance and potential optimality of decision alternatives are presented. In the case of testing potential optimality, the proposed appoach leads to the consideration of a couple of mutually dual linear programming problems. The analysis of these problems gives valuable information for the decision maker. In particular, if a decision alternative is not potentially optimal, then a mixed alternative dominating it is defined by a solution to one of the LP problems. This statement generalizes similar results known for some special cases. The interpretation of the mixed alternative is discussed and compared to its analogue in a data envelopment analysis context.     

Classifying efficient alternatives in SMAA using cross confidence factors

Stochastic multicriteria acceptability analysis (SMAA) is a family of methods for aiding multicriteria group decision making. These methods are based on exploring the weight space in order to describe the preferences that make each alternative the most preferred one. The main results of the analysis are rank acceptability indices, central weight vectors and confidence factors for different alternatives. The rank acceptability indices describe the variety of different preferences resulting in a certain rank for an alternative; the central weight vectors represent the typical preferences favouring each alternative; and the confidence factors measure whether the criteria data are sufficiently accurate for making an informed decision.In some cases, when the problem involves a large number of efficient alternatives, the analysis may fail to discriminate between them. This situation is revealed by low confidence factors. In this paper we develop cross confidence factors, which are based on computing confidence factors for alternatives using each other’s central weight vectors. The cross confidence factors can be used for classifying efficient alternatives into sets of similar and competing alternatives. These sets are related to the concept of reference sets in Data Envelopment Analysis (DEA), but generalized for stochastic models. Forming these sets is useful when trying to identify one or more most preferred alternatives, or suitable compromise alternatives. The reference sets can also be used for evaluating whether criteria need to be measured more accurately, and at which alternatives the measurements should be focused. This may cause considerable savings in measurement costs. We demonstrate the use of the cross confidence factors and reference sets using a real-life example.