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Rough Sets in Decision Making

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Scientific analysis of decision problems aims at giving the decision maker (DM) a recommendation concerning a set of objects (also called alternatives, solutions, acts, actions, options, candidates,…) evaluated from multiple points of view considered relevant for the problem at hand and called attributes(also called features, variables, criteria, objectives, …). For example, a decision canconcern:

  1. 1)

    diagnosis of pathologies for a set of patients, where patients are objects of thedecision, and symptoms and results of medical tests are the attributes,

  2. 2)

    assignment of enterprises to classes of risk, where enterprises are objects of thedecision, and financial ratio indices and other economic indicators, such as the market structure,the technology used by the enterprise and the quality of management, are the attributes,

  3. 3)

    selection of a car to be bought from among a given set of cars, where cars are objectsof the decision, and maximum speed, acceleration, price,...

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Abbreviations

Multiple attribute (or multiple criteria) decision support:

aims at giving the decision maker(DM) a recommendation concerning a set of objects A (also called alternatives, actions,acts, solutions, options, candidates, …) evaluated from multiple points of view calledattributes (also called features, variables, criteria, objectives, …).

Main categories of multiple attribute (or multiple criteria) decision problems:

  • classification, when the decision aims at assigning each object to one of predefined classes,

  • choice, when the decision aims at selecting the best object,

  • ranking, when the decision aims at ordering objects from the best to the worst.

Two kinds of classification problems are distinguished:

  • taxonomy, when the value sets of attributes and the predefined classes are notpreference ordered,

  • ordinal classification (also known asmultiple criteria sorting), when the value sets of attributes and the predefined classes are preferenceordered.

Two kinds of choice problems are distinguished:

  • discrete choice, when the set of objects is finite and reasonably small to belisted,

  • multiple objective optimization, when the set of objects is infinite and defined byconstraints of a mathematical program.

If value sets of attributes are preference-ordered, they are called criteria orobjectives, otherwise they keep the name of attributes.

Criterion:

is a real-valued function g i defined on A, reflecting a worth of objectsfrom a particular point of view, such that in order to compare any two objects \( { a,b \in A } \) fromthis point of view it is sufficient to compare two values: \( { g_i(a) } \) and \( { g_i(b) } \).

Dominance:

Object  a is non-dominated in set A (Pareto-optimal) if and only if there is no otherobject b in A such that b is not worsethan a on all considered criteria, and strictly better on at least one criterion.

Preference model:

is a representation ofa value system of the decision maker on the set of objects with vector evaluations.

Decision under uncertainty:

takes into account consequences of decisions that distribute overmultiple states of nature with given probability. The preference order, characteristic for datadescribing multiple attribute decision problems, concerns also decision under uncertainty,where the objects correspond to acts, attributes are outcomes (gain or loss) to be obtained with a given probability, and the problemconsists in ordinal classification, choice, or ranking of the acts.

Rough set:

in universe U is an approximation of a set based on available information aboutobjects of U. The rough approximation is composed of two ordinary sets called lower andupper approximation. Lower approximation is a maximal subset of objects which, according to theavailable information, certainly belong to the approximated set, and upper approximation isa minimal subset of objects which, according to the available information, possibly belong to theapproximated set. The difference between upper and lower approximation is called boundary.

Decision rule:

is a logical statement of the type “if …, then …”, kursiv wherethe premise (condition part) specifies values assumed by one or more condition attributes and theconclusion (decision part) specifies an overall judgment.

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Author information

Authors and Affiliations

  1. Institute of Computing Science, Poznan University of Technology, Poznan, Poland

    Roman Slowinski

  2. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Roman Slowinski

  3. Faculty of Economics, University of Catania, Catania, Italy

    Salvatore Greco & Benedetto Matarazzo

Authors
  1. Roman Slowinski
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  2. Salvatore Greco
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  3. Benedetto Matarazzo
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Slowinski, R., Greco, S., Matarazzo, B. (2009). Rough Sets in Decision Making. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_460

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