Modeling uncertainty in multi-criteria decision analysis

his paper provides a review of multiple criteria decision analysis (MCDA) for cases where attribute evaluations are uncertain. The main aim is to identify different tools which can be used to represent uncertain evaluations, and to broadly survey the available decision models that can be used to support uncertain decision making. The review includes models using probabilities or probability-like quantities; explicit risk measures such as quantiles and variances; fuzzy numbers, and scenarios. The practical assessment of uncertain outcomes and preferences associated with these outcomes is also discussed.     

Multiple criteria aggregation procedure for mixed evaluations

Most of the existing multiple criteria decision-making methods handle one kind of the information imperfections at the same time. Stochastic methods and fuzzy methods constitute typical examples of these methods. However, several multiple criteria modelizations include simultaneously many kinds of the information imperfections. In this work, we propose a multiple criteria aggregation procedure which accepts mixed evaluations, i.e. evaluations which contain different natures of imperfections. It is based on an adaptation of the stochastic dominance results.     

A method for stochastic multiple criteria decision making based on pairwise comparisons of alternatives with random evaluations

This paper proposes a method for solving stochastic multiple criteria decision making (MCDM) problems, where evaluations of alternatives on considered criteria are random variables with known probability density functions or probability mass functions. Probabilities on all possible results of pairwise comparisons of alternatives are first calculated using Probability Theory. Then, all possible results of pairwise comparisons are classified into superior, indifferent and inferior ones using a predefined identification rule. Consequently, the probabilities on all possible results of pairwise comparisons are partitioned into superior, indifferent and inferior probabilities. Furthermore, based on the derived probabilities, an algorithm is developed to rank the alternatives. Finally, a numerical example is used to illustrate the feasibility and validity of the proposed method.     

Notes on ‘Hit-And-Run enables efficient weight generation for simulation-based multiple criteria decision analysis’

In our previous work published in this journal, we showed how the Hit-And-Run (HAR) procedure enables efficient sampling of criteria weights from a space formed by restricting a simplex with arbitrary linear inequality constraints. In this short communication, we note that the method for generating a basis of the sampling space can be generalized to also handle arbitrary linear equality constraints. This enables the application of HAR to sampling spaces that do not coincide with the simplex, thereby allowing the combined use of imprecise and precise preference statements. In addition, it has come to our attention that one of the methods we proposed for generating a starting point for the Markov chain was flawed. To correct this, we provide an alternative method that is guaranteed to produce a starting point that lies within the interior of the sampling space.     

Robustness in operational research and decision aiding: A multi-faceted issue

In Introduction, I explain the meaning I give to the qualifier term ”robust” and justify my preference for the expression robustness concern rather than robustness analysis, which I feel is likely to be interpreted too narrowly. In Section 2, I discuss this concern in more details and I try to clarify the numerous raisons d’être of this concern. As a means of examining the multiple facets of robustness concern more comprehensively, I explore the existing research about robustness, attempting to highlight what I see as the three different territories covered by these studies (Section 3). In Section 4, I refer to these territories to illustrate how responses to robustness concern could be even more varied than they currently are. In this perspective, I propose in Section 5 three new measures of robustness. In the last section, I identify several aspects of the problem that should be examined more closely because they could lead to new avenues of research, which could in turn yield new and innovative responses.     

Regularized estimation for preference disaggregation in multiple criteria decision making

Disaggregation methods have been extensively used in multiple criteria decision making to infer preferential information from reference examples, using linear programming techniques. This paper proposes simple extensions of existing formulations, based on the concept of regularization which has been introduced within the context of the statistical learning theory. The properties of the resulting new formulations are analyzed for both ranking and classification problems and experimental results are presented demonstrating the improved performance of the proposed formulations over the ones traditionally used in preference disaggregation analysis.     

Aspiration level approach in stochastic MCDM problems

The paper considers a discrete stochastic multiple criteria decision making problem. This problem is defined by a finite set of actions A, a set of attributes X and a set of evaluations of actions with respect to attributes E. In stochastic case the evaluation of each action with respect to each attribute takes form of a probability distribution. Thus, the comparison of two actions leads to the comparison of two vectors of probability distributions. In the paper a new procedure for solving this problem is proposed. It is based on three concepts: stochastic dominance, interactive approach, and preference threshold. The idea of the procedure comes from the interactive multiple objective goal programming approach. The set of actions is progressively reduced as the decision maker specifies additional requirements. At the beginning the decision maker is asked to define preference threshold for each attribute. Next, at each iteration the decision maker is confronted with the set of considered actions. If the decision maker is able to make a final choice then the procedure ends, otherwise he/she is asked to specify aspiration level. A didactical example is presented to illustrate the proposed technique.     

An experimental study of the effect of uncertainty representation on decision making

This paper presents the results of an experiment investigating the effects of using different formats for representing uncertain attribute evaluations on decision making. Study participants make a series of hypothetical choices using six uncertainty formats - probability distributions, expected values, standard deviations, three-point (minimum-median-maximum) approximations, quantiles, and scenarios - and effects on decision making are tracked in terms of the quality of the final choice, the specific characteristics of the selected alternatives, and the difficulty experienced in making a decision. The results provide insights into how subjects make single- and multi-criteria choices in the presence of uncertainty (and some format for representing uncertainty) but in the absence of any real facilitation. The use of probability distributions appeared to overload subjects with information, leading to poorer and more difficult choices than if some intermediate level of summary was used - in particular three-point approximations or quantiles.     

The extended QUALIFLEX method for multiple criteria decision analysis based on interval type-2 fuzzy sets and applications to medical decision making

QUALIFLEX, a generalization of Jacquet-Lagreze’s permutation method, is a useful outranking method in decision analysis because of its flexibility with respect to cardinal and ordinal information. This paper develops an extended QUALIFLEX method for handling multiple criteria decision-making problems in the context of interval type-2 fuzzy sets. Interval type-2 fuzzy sets contain membership values that are crisp intervals, which are the most widely used of the higher order fuzzy sets because of their relative simplicity. Using the linguistic rating system converted into interval type-2 trapezoidal fuzzy numbers, the extended QUALIFLEX method investigates all possible permutations of the alternatives with respect to the level of concordance of the complete preference order. Based on a signed distance-based approach, this paper proposes the concordance/discordance index, the weighted concordance/discordance index, and the comprehensive concordance/discordance index as evaluative criteria of the chosen hypothesis for ranking the alternatives. The feasibility and applicability of the proposed methods are illustrated by a medical decision-making problem concerning acute inflammatory demyelinating disease, and a comparative analysis with another outranking approach is conducted to validate the effectiveness of the proposed methodology.     

Range-based Multi-Actor Multi-Criteria Analysis: A combined method of Multi-Actor Multi-Criteria Analysis and Monte Carlo simulation to support participatory decision making under uncertainty

Concerns about environmental and social effects have made Multi-Criteria Decision Making (MCDM) increasingly popular. Decision making in complex contexts often – possibly always – requires addressing an aggregation of multiple issues to meet social, economic, legal, technical, and environmental objectives. These values at stake may affect different stakeholders through distributional effects characterized by a high and heterogeneous uncertainty that no social actors can completely control or understand. On this basis, we present a new process framework that aims to support participatory decision making under uncertainty: the range-based Multi-Actor Multi-Criteria Analysis (range-based MAMCA). On the one hand, the process framework explicitly considers stakeholders’ objectives at an output level of aggregation. On the other hand, by means of a Monte Carlo analysis, the method also provides an exploratory scenario approach that enables the capture of the uncertainty, which stems from the complex context evolution. Range-based MAMCA offers a unique participatory process framework that enables us (1) to identify the alternatives pros and cons for each stakeholder group; (2) to provide probabilities about the risk of supporting mistaken, or at least ill-suited, decisions because of the uncertainty regarding to the decision-making context; (3) to take the decision-makers’ limited control of the actual policy effects over the implementation of one or several options into account. The range-based MAMCA framework is illustrated by means of our first case study that aimed to assess French stakeholders’ support for different biofuel options by 2030.     

Methodological aspects of multi-criteria decision analysis for policy support: A case study on pharmaceutical removal from hospital wastewater

Decision making in public and political contexts can be complex. Multi-attribute value/utility theory (MAVT/MAUT) can support such decision processes by providing a transparent framework that helps focusing on objectives and corresponding degrees of achievement by different alternatives.     

On the maximum number of feasible ranking sequences in multi-criteria decision making problems

In many decision problems the focus is on ranking a set of m alternatives in terms of a number, say n, of decision criteria. Given are the performance values of the alternatives for each one of the criteria and the weights of importance of the criteria. This paper demonstrates that if one assumes that the criteria weights are changeable, then the number of all possible rankings may be significantly less than the upper limit of m!. As a matter of fact, this paper demonstrates that the number of possible rankings is a function of the number of alternatives and the number of criteria. These findings are important from a sensitivity analysis point of view or when a group decision making environment is considered.     

Modelling impacts of cropping systems: Demands and solutions for DEX methodology

Decision modelling of diverse groups of problems makes different requirements to the modelling methodologies and software. We present an actual decision problem and the required characteristics of corresponding decision models. The problem is from agronomy and addresses the ecological and economic impacts of cropping systems, with the focus on the differences between cropping systems with conventional crops and the ones with genetically modified crops. We describe the extensions of an existing DEX qualitative multi-attribute modelling methodology, which were made to cope with the challenges of the problem. The extensions address general hierarchical structures, probabilistic utility functions and numerical values of basic attributes. A new, freely available software tool called proDEX was implemented to support the extended methodology. In this paper we describe the problem of cropping system assessment, propose methodological extensions to DEX, and present the implementation of proDEX.     

Multi-criteria decision aid for the formulation of sustainable technological energy priorities using linguistic variables

Implementation of new and innovative energy technologies is a key mean towards a sustainable energy system. Currently, governments have to decide from an increasingly diverse mix of them, the ones which warrant support, including funding and other incentives for private sector efforts. However, appraising energy technologies in terms of their sustainability is a really complex task, considering the series of uncertainties and implications that have to be encountered so as to obtain realistic and transparent results. In this context, the main aim of this paper is to present a direct and flexible multi-criteria decision making approach, using linguistic variables, to assist policy makers in formulating sustainable technological energy priorities. Furthermore, its software realization will be applied to a number of technologies, in the context of the Greek Technology Foresight Programme, and the results will be presented and discussed.     

The nearest adjoining order method for pairwise comparisons in the form of difference of ranks

The problem of ranking of elements from some finite set on the basis of nearest adjoining order method for pairwise comparisons is investigated in this paper. It is assumed that in the set under consideration there exists a weak preference relation, which is to be identified (estimated) on the basis of pairwise comparisons in the form of difference of ranks. Moreover, the results of comparisons may be disturbed with random errors; the assumptions made about error distributions are not restrictive. The paper comprises: the problem formulation (definitions, assumptions, and optimisation problem, which provides the NAO solution) and the theoretical background – the form of distributions of random variables which make it possible to determine the properties of NAO solution, in particular, evaluation of the probability, that the NAO solution is equivalent to the errorless one. The approach presented in the paper can be extended to the case of more than one comparison for each pair of elements, i.e., completely formalised multi-experts ranking procedure. This revised version was published online in June 2006 with corrections to the Cover Date.     

Hit-And-Run enables efficient weight generation for simulation-based multiple criteria decision analysis

Models for Multiple Criteria Decision Analysis (MCDA) often separate per-criterion attractiveness evaluation from weighted aggregation of these evaluations across the different criteria. In simulation-based MCDA methods, such as Stochastic Multicriteria Acceptability Analysis, uncertainty in the weights is modeled through a uniform distribution on the feasible weight space defined by a set of linear constraints. Efficient sampling methods have been proposed for special cases, such as the unconstrained weight space or complete ordering of the weights. However, no efficient methods are available for other constraints such as imprecise trade-off ratios, and specialized sampling methods do not allow for flexibility in combining the different constraint types. In this paper, we explore how the Hit-And-Run sampler can be applied as a general approach for sampling from the convex weight space that results from an arbitrary combination of linear weight constraints. We present a technique for transforming the weight space to enable application of Hit-And-Run, and evaluate the sampler’s efficiency through computational tests. Our results show that the thinning factor required to obtain uniform samples can be expressed as a function of the number of criteria n as φ(n) = (n − 1)³. We also find that the technique is reasonably fast with problem sizes encountered in practice and that autocorrelation is an appropriate convergence metric.     

A new distance measure including the weak preference relation: Application to the multiple criteria aggregation procedure for mixed evaluations

We introduce a new distance measure between two preorders that captures indifference, strict preference, weak preference and incomparability relations. This measure is the first to capture weak preference relations. We illustrate how this distance measure affords decision makers greater modeling power to capture their preferences, or uncertainty and ambiguity around them, by using our proposed distance measure in a multiple criteria aggregation procedure for mixed evaluations.     

A new data envelopment analysis method for priority determination and group decision making in the analytic hierarchy process

The DEAHP method for weight deviation and aggregation in the analytic hierarchy process (AHP) has been found flawed and sometimes produces counterintuitive priority vectors for inconsistent pairwise comparison matrices, which makes its application very restrictive. This paper proposes a new data envelopment analysis (DEA) method for priority determination in the AHP and extends it to the group AHP situation. In this new DEA methodology, two specially constructed DEA models that differ from the DEAHP model are used to derive the best local priorities from a pairwise comparison matrix or a group of pairwise comparison matrices no matter whether they are perfectly consistent or inconsistent. The new DEA method produces true weights for perfectly consistent pairwise comparison matrices and the best local priorities that are logical and consistent with decision makers (DMs)’ subjective judgments for inconsistent pairwise comparison matrices. In hierarchical structures, the new DEA method utilizes the simple additive weighting (SAW) method for aggregation of the best local priorities without the need of normalization. Numerical examples are examined throughout the paper to show the advantages of the new DEA methodology and its potential applications in both the AHP and group decision making.     

Post-objective determination of weights of the evaluation factors in public procurement tenders

In public procurement tenders the awarding criterion of the most economically advantageous bid employs weights to aggregate the numerical scores assigned to each proposal with respect to different evaluation factors. Typically these weights are fixed and subjectively set in advance. Methods, which objectively determine the weights after the opening of the sealed bids on the basis of the most or least favorable weights for each proposal, are developed. Post-objective methods of weight determination are shown to enhance the integrity of the evaluation process and to limit corruption in a public tender. The connection of Data Envelopment Analysis, which has been extensively applied to measure supplier efficiency, with the developed methods, is explored. Average least and most favorable weights are derived and optimal bidding strategies in this setting are presented.     

The hierarchical and social decision making procedure for bi-objective optimization problems by using the simple majority decision rule

Multi-objective optimization has been successfully applied to problems of industrial design, problems of quality control and production management, and problems of finance. The theme of these applications is how to choose the best solution for the decision makers out of a set of non-inferior solutions to a multi-objective optimization problem. For this purpose, an optimization model with hierarchical structure, whose lower problem is a multi-objective optimization problem and the upper problem is a preference optimization problem on a set of non-inferior solutions, must be constructed. This kind of hierarchical problems have been previously analyzed only with regard to linear programming problems by Benson[6]. In this paper, an algorithm is derived that provides a solution as a social choice, obtained by aggregating plural decision-makers' preferences. In the case of the simple majority rule, the bi-objective problem is transformed into an -parameter choice problem, and the golden section method is applied. The availability of the approach is demonstrated with the means of an illustrative example.Technische Universität BerlinFaculty of Science and Technology, Keio University