Multicriteria Models for Learning Ordinal Data: A Literature Review <s> Criteria Weighting <s> Abstract We suggest a method for providing descriptive information about the acceptability of decision alternatives in discrete co-operative group decision-making problems. The new SMAA-O method is a variant of the stochastic multicriteria acceptability analysis (SMAA). SMAA-O is designed for problems where criteria information for some or all criteria is ordinal; that is, experts (or decision-makers) have ranked the alternatives according to each (ordinal) criterion. Considerable savings can be obtained if rank information for some or all the criteria is sufficient for making decisions without significant loss of quality. The approach is particularly useful for group decision making when the group can agree on the use of an additive decision model but only partial preference information, or none at all, is available. <s> BIB001 </s> Multicriteria Models for Learning Ordinal Data: A Literature Review <s> Criteria Weighting <s> Stochastic multicriteria acceptability analysis (SMAA) comprises a family of multiple criteria decision-aiding (MCDA) methods for problems including incomplete, imprecise, and uncertain information. Methods of the family allow solving MCDA problems of various types. Even though the methods have been applied in the past in various real-life decision-making situations, the structure of a unified SMAA framework has not been studied. In this paper we describe the methods of the family and define a unified SMAA framework. We also point out the key points in the methodology for future research. Copyright © 2007 John Wiley & Sons, Ltd. <s> BIB002 </s> Multicriteria Models for Learning Ordinal Data: A Literature Review <s> Criteria Weighting <s> Abstract 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. The computations in SMAA require the evaluation of multidimensional integrals that must in practice be computed numerically. In this paper we present efficient methods for performing the computations through Monte Carlo simulation, analyze the complexity, and assess the accuracy of the presented algorithms. We also test the efficiency of these methods empirically. Based on the tests, the implementation is fast enough to analyze typical-sized discrete problems interactively within seconds. Due to almost linear time complexity, the method is also suitable for analysing very large decision problems, for example, discrete approximations of continuous decision problems. <s> BIB003 </s> Multicriteria Models for Learning Ordinal Data: A Literature Review <s> Criteria Weighting <s> This paper proposes and evaluates a descriptive multiattribute choice model based upon the notion of acceptability developed in the stochastic multicriteria acceptability analysis (SMAA) methods. The acceptability index is simply the relative proportion of all possible preference structures that support the selection of a particular alternative, and is related to a model of choice in which attribute evaluations are relatively stable but what is desired changes as if at random. The model is tested using two longitudinal surveys of FMCG markets in Europe. The acceptability index is found to be positively associated with relative purchase frequencies at the individual and aggregate level, inversely related to defection rates, and positively related to changes in relative purchase frequencies over a six-month period. <s> BIB004 </s> Multicriteria Models for Learning Ordinal Data: A Literature Review <s> Criteria Weighting <s> Several combined heat and power (CHP) system options are presented to assess and compare with respect to the end users' requirements. According to the obtained data from literature, 16 kinds of CHP systems are evaluated using gray relational method in multi-criteria, such as technology, economy, environment, and society. The results depend greatly on the criteria weights in multi-criteria evaluations. Aiming to obtain rational result, this paper reviews the weighting method briefly and proposes an optimal combined weighting method to consider the subjectivity of decision-maker and the objectivity of measurement data. The singular value decomposition aggregation method is employed to verify the rationality of evaluation result. Through multi-criteria evaluation and discussions, the combination weighting method is recommended to be used in the selection of CHP schemes. Finally, the best CHP system is selected and the most conspicuous factors having great impact on CHP system with respect to the users' requirements are given out. Copyright © 2009 John Wiley & Sons, Ltd. <s> BIB005 </s> Multicriteria Models for Learning Ordinal Data: A Literature Review <s> Criteria Weighting <s> Multi-criteria decision analysis (MCDA) methods have become increasingly popular in decision-making for sustainable energy because of the multi-dimensionality of the sustainability goal and the complexity of socio-economic and biophysical systems. This article reviewed the corresponding methods in different stages of multi-criteria decision-making for sustainable energy, i.e., criteria selection, criteria weighting, evaluation, and final aggregation. The criteria of energy supply systems are summarized from technical, economic, environmental and social aspects. The weighting methods of criteria are classified into three categories: subjective weighting, objective weighting and combination weighting methods. Several methods based on weighted sum, priority setting, outranking, fuzzy set methodology and their combinations are employed for energy decision-making. It is observed that the investment cost locates the first place in all evaluation criteria and CO2 emission follows closely because of more focuses on environment protection, equal criteria weights are still the most popular weighting method, analytical hierarchy process is the most popular comprehensive MCDA method, and the aggregation methods are helpful to get the rational result in sustainable energy decision-making. <s> BIB006 </s> Multicriteria Models for Learning Ordinal Data: A Literature Review <s> Criteria Weighting <s> We consider problems where multiple decision makers (DMs) want to choose their most preferred alternative from a finite set based on multiple criteria. Several approaches to support DMs in such problems have been suggested. Prospect theory has appealed to researchers through its descriptive power, but rare attempts have been made to apply it to support multicriteria decision making. The basic idea of prospect theory is that alternatives are evaluated by a difference function in terms of gains and losses with respect to a reference point. The function is suggested to be concave for gains and convex for losses and steeper for losses than for gains. Stochastic multicriteria acceptability analysis (SMAA) is a family of multicriteria decision support methods that allows representing inaccurate, uncertain, or partly missing information about criteria measurements and preferences through probability distributions. SMAA methods are based on exploring the weight and criteria measurement spaces in order to describe weights that would result in a certain rank for an alternative. This paper introduces the SMAA-P method that combines the piecewise linear difference functions of prospect theory with SMAA. SMAA-P computes indices that measure how widely acceptable different alternatives are with assumed behavior. SMAA-P can be used in decision problems, where the DMs' preferences (weights, reference points and coefficients of loss aversion) are difficult to assess accurately. SMAA-P can also be used to measure how robust a decision problem is with respect to preference information. We demonstrate the method by reanalyzing a past real-life example. <s> BIB007
Criteria weighting can be considered one of the most important steps for the decision maker. Once it weights the importance of each criterion, acting as a trade-off between criteria  that will be considered in the decision process, subtle changes can produce different outcome BIB005 . Methods for weighting criteria encompass equal weights, rank-order and hybrid approaches where after some considerations from the DM, weighting can be performed by a subjective or objective method BIB005 BIB006 . Equal weights (w j = 1/d) is not valuable once relative importance among the criteria is ignored. Remains rank-order weighting approaches and their derivations to overcome these limitations. Another issue is that when dealing with uncertainty or incomplete information in any decision problem, the DM may not be reliable to define her/his preferences accurately. One way to handle this type of information is to represent preferences by a suitable distribution using stochastic multicriteria acceptability analysis (SMAA) methods. Several methods have been proposed in the literaturee.g. BIB004 BIB001 BIB007 BIB003 to name a few. SMAA-O proposed in BIB001 was an extension of SMAA works BIB002 BIB003 applied to ordinal (and cardinal) criteria. The problem is that, in the authors approach, an ordered criteria cannot be used directly in MC model. Therefore, it is assumed that exists a cardinal measure that corresponds to the known ordinal criteria and by considering consistent mappings between ordinal and cardinal scales, they randomly simulate such mapping through Monte Carlo iterations. Or in other words, ordinal data is converted into stochastic cardinal data by simulating consistent mappings between ordinal and cardinal scales that preserve the given labels. In SMAA literature review work of Tervonen BIB002 they claim that such simulations are not necessary since cardinal values can be interpreted directly.