Abstract
Ordering and classification of objects by their properties are among the typical problems in multiple criteria decision aiding (MCDA). The difficulties of choice problems increase when the same object may exist in several copies with different attributes’ values, and values of different attributes may be repeated within the object description. For example, such situation arises when several experts estimate alternatives upon multiple criteria. In this case, individual expert assessments may be similar, diverse, or contradictory. Various techniques for classification of alternatives or their ranking have been developed. But most of the methods do not pay a serious consideration to contradictions and inconsistencies in decision makers’ (DM) preferences and a problem description. Group verbal decision analysis (GroupVDA) is a new methodological approach in the MCDA area, which enlarges verbal decision analysis (VDA) approach to a group decision. GroupVDA deals with choice problems where preferences of several decision makers may be discordant, and alternatives are described with manifold repeating quantitative and qualitative attributes. New GroupVDA methods are based on the theory of multisets or sets with repeating elements, and represent multi-attribute objects as points in multiset metric spaces.
Key Terms in this Chapter
Aggregation and Ranking Alternatives nearby the Multi-Attribute Ideal Situations (ARAMIS): ARAMIS is the method for group ordering multi-attribute objects represented as multisets by their closeness to any “ideal” objects in a multiset metric space. The ranking of all objects is found without building many different expert arrangements by many criteria, and without an aggregation them into a common ranking. The object arrangement takes into account various inconsistent and contradictory expert estimates without forcing one to find a compromise among them.
Operations with Multisets: Union A?B, intersection AnB, arithmetic addition A+B, arithmetic subtraction A-B, symmetric difference A?B, complement =Z–A, multiplication by a scalar (reproduction) c•A, arithmetic multiplication ?•?, arithmetic power ??, direct product A×B, direct power (×A)n. In general, arithmetic addition, multiplication by ? scalar, arithmetic multiplication, and raising to an arithmetic power are not defined in the set theory.
Verbal Decision Analysis (VDA): VDA emphasizes ill-structured discrete choice problems, which are represented with quantitative and qualitative attributes. The most important features of VDA are as follows: (1) the problem description with a professional language, which is natural and habitual for a decision maker; (2) a usage of verbal (nominative, ordinal) data on all stages of the problem analysis and solution without transformation into a numerical form; (3) an examination of decision maker’s judgments consistency; (4) a logical and psychological validity of decision rules; and (5) an explanation of intermediate and final results.
Measure of Multiset m: Measure of multiset m is a real-valued non-negative function defined on the algebra of multisets L(Z).
Multiset Metric Space (A,d): Multiset metric space (A,d) is a collection A={A1,...,An} of multisets with any distance d between multisets.
Decision Maker (DM): The DM is a person who is responsible for solution of choice problem.
MASKA (Russian abbreviation for the name: Multi-Attribute Consistent Classification of Alternatives): MASKA is the method for group classifying multi-attribute objects, which are represented as multisets. The method allows us to construct some generalized decision rules for a selection of “good” and “bad” objects from a set of contenders that approximates many inconsistent and contradictory individual sorting rules of several actors with the demanded level of approximation rate.
Group Verbal Decision Analysis (GroupVDA): Group VDA is a new methodological approach in the MCDA area, which enlarges VDA approach to a group decision. GroupVDA deals with choice problems where preferences of several decision makers may be discordant, alternatives are described with manifold repeating quantitative and qualitative attributes, and may exist in several copies.