Domain structuring for knowledge-based multiattribute classification (a verbal decision analysis approach)

Classification problems in decision making are, at least, ill-structured or even unstructured ones, since, among other things, human judgments (i.e., Decision Maker preferences and/or expert knowledge) are the primary sources of information for their solving. Thus, not only the classification rules eliciting, but the application domain structuring as well, is a complex problem itself. The paper focuses on knowledge-based classification problem structuring in the context of complete (up to the expert knowledge) and consistent knowledge base construction for a Diagnostic Decision Support System. Two structuring techniques are proposed as expert aids, as well as an approach to large-size problem decomposition. It is asserted that application domain structuring and classification rules eliciting have to be arranged as interconnected procedures.     

Some interval-valued 2-tuple linguistic aggregation operators and application in multiattribute group decision making

This paper deals with multiattribute group decision making (MAGDM) problems with interval-valued 2-tuple linguistic information. First, we introduce some new aggregation operators, such as the interval-valued 2-tuple weighted geometric (IVTWG) operator, the interval-valued 2-tuple ordered weighted geometric (IVTOWG) operator, the generalized interval-valued 2-tuple weighted average (GIVTWA) operator and the generalized interval-valued 2-tuple ordered weighted average (GIVTOWA). Then, we discuss their desired properties and relationships among them. Furthermore, we put forward a new method to determine the weight vector of interval-valued 2-tuple aggregation operator based on the concept of degree of precision. Finally, a numerical example is provided to illustrate the efficiency of the proposed method in dealing with interval-valued 2-tuple linguistic information under multi-granular linguistic contexts.     

The GOWA operator based approach to multiattribute decision making using intuitionistic fuzzy sets

The aim of this paper is to develop a methodology for solving multiattribute decision making (MADM) problems in which weights of attributes and ratings of alternatives on qualitative and quantitative attributes are expressed with intuitionistic fuzzy sets (IFSs). In this methodology, relative membership/satisfaction and non-membership/non-satisfaction degrees are formulated to construct IFSs for numerical values of alternatives on quantitative attributes. Alternatives on qualitative attributes are evaluated using linguistic variables and semantics which are parameterized by IFSs. Hereby, weights and ratings of alternatives on both qualitative attributes and quantitative attributes may be expressed with IFSs in a unified way. The generalized ordered weighted averaging (GOWA) operator is further extended to the situations in which the argument values are IFSs and thus a methodology is developed to solve MADM problems using IFSs. Validity and applicability of the proposed methodology in this paper are illustrated with a real numerical example.     

On qualitatively consistent,transitive and contradictory judgment matrices emerging from multiattribute decision procedures

Several known and newly introduced classes of positive reciprocal matrices emerging from pairwise comparisons in multiattribute decision problems are studied in the paper. Mainly qualitative features in connection with consistency and inconsistency are considered in order to extend the range of the available analytical methods regarding pairwise comparisons. By using graph representation of positive reciprocal matrices, graph theoretic approach is applied for the argumentation. The applied notions and the theorems developed in the paper can be useful for eliminating the illogical data that may occur during pairwise comparisons.     

Extension of the LINMAP for multiattribute decision making under Atanassov’s intuitionistic fuzzy environment

The aim of this article is further extending the linear programming techniques for multidimensional analysis of preference (LINMAP) to develop a new methodology for solving multiattribute decision making (MADM) problems under Atanassov’s intuitionistic fuzzy (IF) environments. The LINMAP only can deal with MADM problems in crisp environments. However, fuzziness is inherent in decision data and decision making processes. In this methodology, Atanassov’s IF sets are used to describe fuzziness in decision information and decision making processes by means of an Atanassov’s IF decision matrix. A Euclidean distance is proposed to measure the difference between Atanassov’s IF sets. Consistency and inconsistency indices are defined on the basis of preferences between alternatives given by the decision maker. Each alternative is assessed on the basis of its distance to an Atanassov’s IF positive ideal solution (IFPIS) which is unknown a prior. The Atanassov’s IFPIS and the weights of attributes are then estimated using a new linear programming model based upon the consistency and inconsistency indices defined. Finally, the distance of each alternative to the Atanassov’s IFPIS can be calculated to determine the ranking order of all alternatives. A numerical example is examined to demonstrate the implementation process of this methodology. Also it has been proved that the methodology proposed in this article can deal with MADM problems under not only Atanassov’s IF environments but also both fuzzy and crisp environments.     

Baseline value specification and sensitivity analysis in multiattribute project portfolio selection

A key issue in applying multi-attribute project portfolio models is specifying the baseline value – a parameter which defines how valuable not implementing a project is relative to the range of possible project values. In this paper we present novel baseline value specification techniques which admit incomplete preference statements and, unlike existing techniques, make it possible to model problems where the decision maker would prefer to implement a project with the least preferred performance level in each attribute. Furthermore, we develop computational methods for identifying the optimal portfolios and the value-to-cost -based project rankings for all baseline values. We also show how these results can be used to (i) analyze how sensitive project and portfolio decision recommendations are to variations in the baseline value and (ii) provide project decision recommendations in a situation where only incomplete information about the baseline value is available.     

Multicriteria decision analysis in a decision tree

The outcomes at the tips of a decision tree cannot always be represented by a single numerical value on a one-dimensional axis. In many decision problems they are multidimensional, and their performance is expressed in a mixture of verbal statements and physical or monetary values. We propose to use the scores of cardinal methods for multicriteria decision analysis in order to represent the relative performance of the outcomes. Thereafter, we evaluate the chance forks in the tree via the corresponding aggregation procedure: in the Multiplicative AHP via weighted geometric means of the scores and in SMART via weighted arithmetic means. The procedure is based on the idea that the numerical values of verbal quantifiers like somewhat more, more, … do not depend on what we compare, whether it is relative importance or relative likelihood.     

Sensitivity analysis incorporating fuzzy evaluation for scaling constants of multiattribute utility functions

A multiattribute utility function can be represented by a function of single-attribute utility functions if the decision maker’s preference satisfies additive independence or mutually utility independence. Additive independence is a preference condition stronger than mutually utility independence, and the multiattribute utility function is in the additive form if the former condition is satisfied, otherwise it is in the multiplicative form. In this paper, we propose a method for sensitivity analysis of multiattribute utility functions in multiplicative form, taking into account the imprecision of the decision maker’s judgment in the procedures for determining scaling constants (attribute weights).     

Generalized evaluation in decision analysis

The paper presents a discussion on evaluation methods in decision analysis. The presentation begins with the discussion of the expected value rule for selection amongst a number of available courses of action. Then a number of other evaluation rules to either replace or supplement the expected value are presented. They are discussed from a choice rather than preference view. To improve the expected value rule (or any other similar rule), it is suggested that it should be supplemented with other, qualitative rules rather than engaging in further modifications in pursuit of the perfect rule. A characteristic of qualitative rules is that they do not rely on multiplying probabilities and values but treat them as separate numeric entities. Once a rule has been agreed upon, it can be applied to all the alternatives, provided there is a computational procedure for evaluating the alternatives under that rule. Delta dominance is introduced as a unifying concept for many of the dominance rules in current use. Dominance and threshold methods are discussed and the kinship between them is pointed out.     

Decomposition of multiattribute expected-utility functions

This paper integrates and extends the theory of the decomposition of multiattribute expected-utility functions based on utility independence. In a preliminary section, the standard decision model of expected utility is briefly discussed, including the fact that the decision maker's preference forlotteries with two outcomes determines the utility function uniquely. The decomposition possibilities of a utility function are captured by the concept ofautonomous sets of attributes, an affine separability of some kind known as generalized utility independence.Overlapping autonomous sets lead to biaffine-associative, i.e.multiplicative oradditive decompositions. The multiplicative representation shows that autonomy has strongerclosure properties than utility independence, for instance with respect to set-theoretic difference. Autonomy is also a concept with a wider scope since it applies to the decomposition of Boolean functions, games and a number of other topics in combinatorial optimization. This relationship to the well-known theory ofsubstitution decomposition in discrete mathematics also reveals a kind of discrete core behind the decomposition of utility functions. The entirety of autonomous sets can be represented by a compact data structure, the so-calledcomposition tree, which frequently corresponds to a natural hierarchy of attributes. Multiplicative/additive ormulti-affine functions correspond to the hierarchy steps. The known representation of multi-affine functions is shown to be given by aMoebius inversion formula. The entire approach has the advantage that it allows the application of more sophisticated representation methods on a detailed level, whereas it employs onlyfinite set theory andarithmetic on the main levels in the hierarchy.     

Vector Ordinal Optimization

Ordinal optimization is a tool to reduce the computational burden in simulation-based optimization problems. So far, the major effort in this field focuses on single-objective optimization. In this paper, we extend this to multiobjective optimization and develop vector ordinal optimization, which is different from the one introduced in Ref. 1. Alignment probability and ordered performance curve (OPC) are redefined for multiobjective optimization. Our results lead to quantifiable subset selection sizes in the multiobjective case, which supplies guidance in solving practical problems, as demonstrated by the examples in this paper.This paper was supported in part by Army Contract DAAD19-01-1-0610, AFOSR Contract F49620-01-1-0288, and a contract with United Technology Research Center (UTRC). The first author received additional funding from NSF of China Grants 60074012 and 60274011, Ministry of Education (China), and a Tsinghua University (Beijing, China) Fundamental Research Funding Grant, and the NCET program of China.The authors are grateful to and benefited from two rounds of reviews from three anonymous referees.     

A generalized decomposable multiattribute utility function

In this paper a generalized decomposable multiattribute utility function (MAUF) is developed. It is demonstrated that this new MAUF structure is more general than other well-known MAUF structures, such as additive, multiplicative, and multilinear. Therefore, it is more flexible and does not require that the decision maker be consistent with restrictive assumptions such as preferential independence conditions about his/her preferences. We demonstrate that this structure does not require any underlying assumption and hence solves the interdependence among attributes. Hence there is no need for verification of its structure. Several useful extensions and properties for this generalized decomposable MAUF are developed which simplify its structure or assessment. The concept of utility efficiency is developed to identify efficient alternatives when there exists partial information on the scaling constants of an assumed MAUF. It is assumed that the structure (decomposition) of the MAUF is known and the partial information about the scaling constants of the decision maker is in the form of bounds or constraints. For the generalized decomposable structure, linear programming is sufficient to solve all ensuing problems. Some examples are provided.     

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.     

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.     

Possibility and cost in decision analysis

The thesis of this paper is that a practically relevant decision theory must be based on the concept of possibility. As the concept is interpreted here it covers all the obstacles the decision-maker is facing. In many situations the contemplation of possibility is quite as relevant as the usual concentration on utility and probability. There is a traditional economic concept which conforms to an emphasis on obstacles or possibility in decision-making: opportunity cost. The opportunity cost of a decision is the value of the highest valued possible decision which is inhibited as a result of the decision actually taken. However as opportunity cost is usually interpreted decisions are dichotomized as either possible or impossible to perform. It is argued in the paper that this dichotomization is not very realistic. In reality there must be allowed for a continuum of states between the plain impossibility and the complete possibility. This gradual view of possibility could be dealt with if the fuzzy set theory is used. In the paper it is shown how some relevant concepts regarding possibility could be based on fuzziness and how these concepts could be used to analyze practical situations. A method to deal simultaneously with possibility and probability is stated.     

Ordinal ultrafilters versus P-hierarchy

An earlier paper [Starosolski A., P-hierarchy on βω, J. Symbolic Logic, 2008, 73(4), 1202–1214] investigated the relations between ordinal ultrafilters and the so-called P-hierarchy. The present paper focuses on the aspects of characterization of classes of ultrafilters of finite index, existence, generic existence and the Rudin-Keisler-order.