Linear partial information
Encyclopedia
Linear partial information (LPI) is a method of making decisions based on insufficient or fuzzy information
. LPI was introduced in 1970 by Polish - Swiss mathematician Edward Kofler
(1911–2007) to simplify decision
processes. Comparing to other methods
the LPI-fuzziness is algorithm
ically simple and particularly in decision making
, more practically oriented. Instead of an indicator function the decision maker linear
izes any fuzziness by establishing of linear restrictions for fuzzy probability distributions or normalized weights. In the LPI-procedure the decision maker linear
izes any fuzziness instead of applying a membership function. This can be done by establishing stochastic
and non-stochastic LPI-relations. A mixed stochastic and non-stochastic fuzzification is often a basis for the LPI-procedure. By using the LPI-methods any fuzziness in any decision situation can be considered on the base of the linear
fuzzy logic
.
To obtain the maximally warranted expected value
, the decision maker has to choose the strategy
which maximizes the minimal expected value
. This procedure leads to the MaxEmin - Principle and is an extension of the Bernoulli's principle
.
b) The MaxWmin Principle
This principle leads to the maximal guaranteed weight function
, regarding the extreme weights.
c) The Prognostic Decision Principle (PDP)
This principle is based on the prognosis interpretation of strategies under fuzziness.
The idea of fuzzy equilibrium is based on the optimization principles. Therefore the MaxEmin-, MaxGmin- and PDP-stability have to be analyzed. The violation of these principles leads often to wrong predictions and decisions.
of the corresponding fuzzy states or a disturbance set, the fuzzy equilibrium strategy remains the most cautious one, despite of the presence of the fuzziness. Any deviation from this strategy can cause a loss for the decision maker.
Information
Information in its most restricted technical sense is a message or collection of messages that consists of an ordered sequence of symbols, or it is the meaning that can be interpreted from such a message or collection of messages. Information can be recorded or transmitted. It can be recorded as...
. LPI was introduced in 1970 by Polish - Swiss mathematician Edward Kofler
Edward Kofler
Edward Kofler was a mathematician who made important contributions to game theory and fuzzy logic by working out the theory of linear partial information....
(1911–2007) to simplify decision
Decision making
Decision making can be regarded as the mental processes resulting in the selection of a course of action among several alternative scenarios. Every decision making process produces a final choice. The output can be an action or an opinion of choice.- Overview :Human performance in decision terms...
processes. Comparing to other methods
Fuzzy set
Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced simultaneously by Lotfi A. Zadeh and Dieter Klaua in 1965 as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to...
the LPI-fuzziness is algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
ically simple and particularly in decision making
Decision making
Decision making can be regarded as the mental processes resulting in the selection of a course of action among several alternative scenarios. Every decision making process produces a final choice. The output can be an action or an opinion of choice.- Overview :Human performance in decision terms...
, more practically oriented. Instead of an indicator function the decision maker linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
izes any fuzziness by establishing of linear restrictions for fuzzy probability distributions or normalized weights. In the LPI-procedure the decision maker linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
izes any fuzziness instead of applying a membership function. This can be done by establishing stochastic
Stochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...
and non-stochastic LPI-relations. A mixed stochastic and non-stochastic fuzzification is often a basis for the LPI-procedure. By using the LPI-methods any fuzziness in any decision situation can be considered on the base of the linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
fuzzy logic
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
.
Definition
Any Stochastic Partial Information SPI(p), which can be considered as a solution of a linearLinearIn mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
inequality systemSystemSystem is a set of interacting or interdependent components forming an integrated whole....
, is called Linear Partial Information LPI(p) about probability p. It can be considered as an LPI-fuzzification of the probability p corresponding to the concepts of linear fuzzy logic.
Applications
a) The MaxEmin PrincipleTo obtain the maximally warranted expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
, the decision maker has to choose the strategy
Strategy
Strategy, a word of military origin, refers to a plan of action designed to achieve a particular goal. In military usage strategy is distinct from tactics, which are concerned with the conduct of an engagement, while strategy is concerned with how different engagements are linked...
which maximizes the minimal expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
. This procedure leads to the MaxEmin - Principle and is an extension of the Bernoulli's principle
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...
.
b) The MaxWmin Principle
This principle leads to the maximal guaranteed weight function
Weight function
A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more "weight" or influence on the result than other elements in the same set. They occur frequently in statistics and analysis, and are closely related to the concept of a...
, regarding the extreme weights.
c) The Prognostic Decision Principle (PDP)
This principle is based on the prognosis interpretation of strategies under fuzziness.
Fuzzy equilibrium and stability
Despite the fuzziness of information, it is often necessary to choose the optimal, most cautious strategy, for example in economic planning, in conflict situations or in daily decisions. This is impossible without the concept of fuzzy equilibrium. The concept of fuzzy stability is considered as an extension into a time interval, taking into account the corresponding stability area of the decision maker. The more complex is the model, the softer a choice has to be considered.The idea of fuzzy equilibrium is based on the optimization principles. Therefore the MaxEmin-, MaxGmin- and PDP-stability have to be analyzed. The violation of these principles leads often to wrong predictions and decisions.
LPI equilibrium point
Considering a given LPI-decision model, as a convolutionConvolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
of the corresponding fuzzy states or a disturbance set, the fuzzy equilibrium strategy remains the most cautious one, despite of the presence of the fuzziness. Any deviation from this strategy can cause a loss for the decision maker.
External links
- Tools for establishing dominance with linear partial information and attribute hierarchy
- Linear Partial Information with applications
- Linear Partial Information (LPI) with applications to the U.S. economic policy
- Practical decision making with Linear Partial Information (LPI)
- Stochastic programming with fuzzy linear partial information on probability distribution
- One-shot decisions under Linear Partial Information
Selected references
- Edward Kofler - Equilibrium Points, Stability and Regulation in Fuzzy Optimisation Systems under Linear Partial Stochastic Information (LPI), ProceedingsProceedingsIn academia, proceedings are the collection of academic papers that are published in the context of an academic conference. They are usually distributed as printed books either before the conference opens or after the conference has closed. Proceedings contain the contributions made by researchers...
of the International Congress of CyberneticsCyberneticsCybernetics is the interdisciplinary study of the structure of regulatory systems. Cybernetics is closely related to information theory, control theory and systems theory, at least in its first-order form...
and Systems, AFCET, Paris 1984, pp. 233–240 - Edward Kofler - Decision MakingDecision makingDecision making can be regarded as the mental processes resulting in the selection of a course of action among several alternative scenarios. Every decision making process produces a final choice. The output can be an action or an opinion of choice.- Overview :Human performance in decision terms...
under Linear Partial Information. Proceedings of the European Congress EUFIT, Aachen, 1994, p. 891-896. - Edward Kofler - Linear Partial Information with Applications. Proceedings of ISFL 1997 (International SymposiumSymposiumIn ancient Greece, the symposium was a drinking party. Literary works that describe or take place at a symposium include two Socratic dialogues, Plato's Symposium and Xenophon's Symposium, as well as a number of Greek poems such as the elegies of Theognis of Megara...
on Fuzzy LogicFuzzy logicFuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
), Zurich, 1997, p. 235-239. - Edward Kofler – Entscheidungen bei teilweise bekannter Verteilung der Zustände, Zeitschrift für OR, Vol. 18/3, 1974
- Edward Kofler - Extensive Spiele bei unvollständiger Information, in Information in der Wirtschaft, Gesellschaft für Wirtschafts- und Sozialwissenschaften, Band 126, Berlin 1982