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.     

SMAA - Stochastic multiobjective acceptability analysis

Stochastic multiobjective acceptability analysis (SMAA) is a multicriteria decision support technique for multiple decision makers based on exploring the weight space. Inaccurate or uncertain input data can be represented as probability distributions. In SMAA the decision makers need not express their preferences explicitly or implicitly; instead the technique analyses what kind of valuations would make each alternative the preferred one. The method produces for each alternative an acceptability index measuring the variety of different valuations that support that alternative, a central weight vector representing the typical valuations resulting in that decision, and a confidence factor measuring whether the input data is accurate enough for making an informed decision.     

A simulation-based test of stochastic multicriteria acceptability analysis using achievement functions

Stochastic multicriteria acceptability analysis using achievement functions (SMAA-A) is a preference model for discrete-choice decision making that inverts the traditional goal programming process by asking what combinations of aspirations are necessary to make each alternative the preferred one, rather than what alternative is preferred given a set of aspirations. In this paper, we test the ability of the model to discern good-performing alternatives from poorly-performing ones using a simulation study. Simulation results show that a suitably detailed construction of the acceptability index is particularly important, and that the resulting model can be fruitfully applied in the selection of a shortlist of alternatives from a larger set with only very limited decision maker involvement.     

Restricted domains,arrow-social welfare functions and noncorruptible and non-manipulable social choice correspondences: The case of private alternatives

An n-person social choice problem is considered in which the alternatives are n dimensional vectors, with the ith component of such a vector being the part of the alternatives affecting individual i alone. Assuming that individuals are selfish (individual i must be indifferent between any two alternatives with the same components), that they may be indifferent among alternatives and that each individual may choose his preferences out of a different set of permissible preferences, we prove that any set of restricted domains of preferences admits an n person non-dictatorial Arrow-type social welfare function if and only if it admits a two-person Arrow-type social welfare function: we characterize all the sets of restricted domains of preferences which admit two-person Arrow-type social welfare functions (and therefore also admit n-person Arrow-type social welfare functions) and then we prove that we also characterized all the sets of restricted domains of preferences which admit nondictatorial, nonmanipulable, noncorruptible and rational social choice correspondences.     

Ranking DMUs by using interval DEA cross efficiency matrix with acceptability analysis

Cross efficiency method is an extension of data envelopment analysis (DEA), and has been widely used for ranking performance of decision making units (DMUs). To eliminate the non-uniqueness of cross efficiency scores, the aggressive and benevolent strategies have been proposed as secondary goals to determine the unique cross efficiency score. The current paper aims to propose an alternative strategy which does not consider the preference of the decision maker in choosing aggressive or benevolent strategy. Instead, the paper considers all possible weight sets in weight space when computing the cross efficiency and each DMU is given an interval cross efficiency. By using the stochastic multicriteria acceptability analysis (SMAA-2) method, all DMUs in the interval cross efficiency matrix (CEM) could be fully ranked according to the acceptability indices. A numerical example about efficiency evaluation to seven academic departments in a university is illustrated.     

Single valued neutrosophic cross-entropy for multicriteria decision making problems

A single valued neutrosophic set (SVNS) is an instance of a neutrosophic set, which give us an additional possibility to represent uncertainty, imprecise, incomplete, and inconsistent information which exist in real world. It would be more suitable to apply indeterminate information and inconsistent information measures. In this paper, the cross entropy of SVNSs, called single valued neutrosophic cross entropy, is proposed as an extension of the cross entropy of fuzzy sets. Then, a multicriteria decision-making method based on the proposed single valued neutrosophic cross entropy is established in which criteria values for alternatives are SVNSs. In decision making process, we utilize the single-valued neutrosophic weighted cross entropy between the ideal alternative and an alternative to rank the alternatives corresponding to the cross entropy values and to select the most desirable one(s). Finally, a practical example of the choosing problem of suppliers is provided to illustrate the application of the developed approach.     

Decision making in multiobjective optimization aided by the multicriteria tournament decision method

This paper proposes a new method for multicriteria analysis, named Multicriteria Tournament Decision (MTD). It provides the ranking of alternatives from best to worst, according to the preferences of a human decision-maker (DM). It has some positive aspects such as: it has a simple algorithm with intuitive appeal; it involves few input parameters (just the importance weight of each criterion).The helpfulness of MTD is demonstrated by using it to select the final solution of multiobjective optimization problems in an a posteriori decision making approach. Having at hand a discrete approximation of the Pareto front (provided by a multiobjective evolutionary search algorithm), the choice of the preferred Pareto-optimal solution is performed using MTD.A simple method, named Gain Analysis method (GAM), for verifying the existence of a better solution (a solution associated to higher marginal rates of return) than the one originally chosen by the DM, is also introduced here. The usefulness of MTD and GAM methods is confirmed by the suitable results shown in this paper.     

Decision making in multiobjective optimization aided by the multicriteria tournament decision method

This paper proposes a new method for multicriteria analysis, named Multicriteria Tournament Decision (MTD). It provides the ranking of alternatives from best to worst, according to the preferences of a human decision-maker (DM). It has some positive aspects such as: it has a simple algorithm with intuitive appeal; it involves few input parameters (just the importance weight of each criterion).The helpfulness of MTD is demonstrated by using it to select the final solution of multiobjective optimization problems in an a posteriori decision making approach. Having at hand a discrete approximation of the Pareto front (provided by a multiobjective evolutionary search algorithm), the choice of the preferred Pareto-optimal solution is performed using MTD.A simple method, named Gain Analysis method (GAM), for verifying the existence of a better solution (a solution associated to higher marginal rates of return) than the one originally chosen by the DM, is also introduced here. The usefulness of MTD and GAM methods is confirmed by the suitable results shown in this paper.     

A portfolio model for performance assessment: the <Emphasis Type="Italic">Financial Times</Emphasis> MBA ranking

The ranking of MBA programmes by newspapers and magazines is common and usually controversial. This paper discusses the use of the most popular method of making these rankings via a multicriteria model which uses the weighted sum of a number of performance measures to give an overall score on which selection or ranking may be based. The weights are a quantitative model of the preferences of those making the evaluation. Many methods are available to obtain weights from preference statements so that for any set of preferences a number of different weight sets can be found depending on the method used. Cognitive limits lead to inconsistency in preference judgements so that weights may be subject both to uncertainty and to bias. It is proposed that choosing weights to minimize discrimination between alternatives (not weights) guards against unjustified discrimination between alternatives. Applying the method to data collected by the Financial Times shows the effect of varying the level of discrimination between weights and also the effect of using a reduced data set made necessary by the partial publication of information.     

Ranking alternatives using fuzzy numbers

We investigate the problem of employing expert opinion to rank alternatives across a set of criteria. The experts use fuzzy numbers to express their preferences and we employ fuzzy arithmetic to compute an issue's fuzzy ranking. This leads to a partition of the alternatives into sets H₁, H₂,… where H₁ contains the highest ranked issues, H₂ has all the second highest ranked alternatives, etc. The total ranking process is shown to possess a number of important properties. An example is presented to illustrate the method.     

When conflict induces the expression of incomplete preferences

Multicriteria conflict arises in pairwise comparisons, where each alternative outperforms the other one on some criterion, which imposes a trade-off. Comparing two alternatives can be difficult if their respective advantages are of high magnitude (the attribute spread is large). In this paper, we investigate to which extent conflict in a comparison situation can lead decision makers to express incomplete preferences, that is, to refuse to compare the two alternatives, or to be unable to compare them with confidence. We report on an experiment in which subjects expressed preferences on pairs of alternatives involving varying conflicts. Results show that depending on whether the participants are allowed to express incomplete preferences or not, attribute spread has a different effect: a large attribute spread increases the frequency of incomparability statements, when available, while it increases the use of indifference statements when only indifference and preference answers are permitted. These results lead us to derive some implications for preference elicitation methods involving comparison tasks.     

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?     

Locating a waste treatment facility by using stochastic multicriteria acceptability analysis with ordinal criteria

In this paper we describe a real-life application of an ordinal multicriteria method in the context of choosing a location for a waste treatment facility near Lappeenranta in South-Eastern Finland. The associated environmental impact assessment (EIA) procedure is briefly described. The application was characterized by two interesting properties: no preference information was available, and only ordinal measurements for the criteria were available. The large amount of data obtained was then analyzed using the SMAA-O method – Stochastic Multicriteria Acceptability Analysis with Ordinal criteria designed for this problem setting. SMAA-O converts ordinal information into cardinal data by simulating all possible mappings between ordinal and cardinal scales that preserve the given rankings. As with the basic SMAA-method, the decision makers' (DMs) unknown or partly known preferences are at the same time simulated by choosing weights randomly from appropriate distributions. The main results of the analysis are acceptability indices for alternatives describing the variety of preferences that could make each alternative the best choice. Based on these and additional considerations, the DMs made the final choice for the location of the plant.     

Vectors of indicators and pointer function in the Multistage Bipolar Method

The Multistage Bipolar Method considered in the paper deals with multistage decision processes. Multistage alternatives are not compared directly to each other, but they are confronted with the stage sets of reference objects—desirable and non-acceptable. In the paper vectors and pointer functions are defined. The aim of the paper is to apply them to classify and rank multistage alternatives and search for the final solution. Our method simplifies the procedure of finding the final solution and allows to use single criterion dynamic programming to solve the problem.

     

Systemic decision making in AHP: a Bayesian approach

Systemic decision making is a new approach for dealing with complex multiactor decision making problems in which the actors’ individual preferences on a fixed set of alternatives are incorporated in a holistic view in accordance with the “principle of tolerance”. The new approach integrates all the preferences, even if they are encapsulated in different individual theoretical models or approaches; the only requirement is that they must be expressed as some kind of probability distribution. In this paper, assuming the analytic hierarchy process (AHP) is the multicriteria technique employed to rank alternatives, the authors present a new methodology based on a Bayesian analysis for dealing with AHP systemic decision making in a local context (a single criterion). The approach integrates the individual visions of reality into a collective one by means of a tolerance distribution, which is defined as the weighted geometric mean of the individual preferences expressed as probability distributions. A mathematical justification of this distribution, a study of its statistical properties and a Monte Carlo method for drawing samples are also provided. The paper further presents a number of decisional tools for the evaluation of the acceptance of the tolerance distribution, the construction of tolerance paths that increase representativeness and the extraction of the relevant knowledge of the subjacent multiactor decisional process from a cognitive perspective. Finally, the proposed methodology is applied to the AHP-multiplicative model with lognormal errors and a case study related to a real-life experience in local participatory budgets for the Zaragoza City Council (Spain).     

A rank-dependent bi-criterion equilibrium model for stochastic transportation environment

The paper proposes a rank-dependent bi-criterion (travel time & monetary travel cost) equilibrium model for route choice problems, stochasticities in both the criteria measurements and the subjective preferences are considered simultaneously. Travelers rank all the choices, according to the generalized travel dis-utility, then choose from the first several (see K  ) best ranked ones. By searching inversely the supporting preference sets for each alternative in each rank, the overall choice probability of a path is determined. The equilibrium model is formulated and transformed into a fixed-point problem. The existence of the equilibrium is given out for a simple two-link network, however may not be guaranteed for more complex network topologies. When K=1$K=1$, the proposed model reduces to the optimal user equilibrium that allows for the stochasticities of criteria measurements and the arbitrarily distributed preferences. Some remarks about the selection of some parameters in the new model are discussed and also the solution algorithms. Two numerical examples are presented to illustrate the implementation of the model, and also the capability and flexibility of the new model in handling the heterogeneity in traveler preferences and requirements. The paper concludes with discussions about the assumptions and limitations of the new model and possible future research opportunities as well.     

Multicriteria fuzzy decision-making method using entropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy sets

A multicriteria fuzzy decision-making method based on weighted correlation coefficients using entropy weights is proposed under interval-valued intuitionistic fuzzy environment for the some situations where the information about criteria weights for alternatives is completely unknown. To determine the entropy weights with respect to a decision matrix provided as interval-valued intuitionistic fuzzy sets (IVIFSs), we propose two entropy measures for IVIFSs and establish an entropy weight model, which can be used to determine the criteria weights on alternatives, and then propose an evaluation formula of weighted correlation coefficient between an alternative and the ideal alternative. The alternatives can be ranked and the most desirable one(s) can be selected according to the values of the weighted correlation coefficients. Finally, two applied examples demonstrate the applicability and benefit of the proposed method: it is capable for handling the multicriteria fuzzy decision-making problems with completely unknown weights for criteria.     

Stochastic multicriteria acceptability analysis using the data envelopment model

Data envelopment analysis (DEA) and stochastic multicriteria acceptability analysis (SMAA-2) are methods for evaluating alternatives based on multiple criteria. While DEA is mainly an ex-post tool used for classifying alternatives into efficient and inefficient ones, SMAA-2 is an ex-ante tool for supporting multiple criteria decision-making. Both methods use a kind of value function where the importance of criteria is modeled using weights. Unlike many other methods, neither DEA nor SMAA-2 requires decision-makers’ weights as input. Instead, these so-called non-parametric methods explore the weight space in order to identify weights favorable for each alternative. This paper introduces the SMAA-D method, which is a combination of DEA and SMAA-2. SMAA-D can be characterized as an extension of DEA to handle uncertain or imprecise data to provide stochastic efficiency measures. Alternatively, the combined method can be seen as a variant of SMAA-2 with a DEA-type value function.