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. Compared to
other methods the LPI-fuzziness is
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
ically simple and particularly in
decision making
In psychology, decision-making (also spelled decision making and decisionmaking) is regarded as the cognitive process resulting in the selection of a belief or a course of action among several possible alternative options. It could be either rati ...
, more practically oriented. Instead of an
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
the decision maker
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
izes any fuzziness by establishing of linear restrictions for fuzzy probability distributions or normalized weights. In the LPI-procedure the decision maker
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
izes any fuzziness instead of applying a membership function. This can be done by establishing
stochastic
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
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
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
fuzzy logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
.
Definition
Any Stochastic Partial Information SPI(p), which can be considered as a solution of a
linear inequality system, 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
;The MaxEmin Principle
: To obtain the maximally warranted
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
, the decision maker has to choose the
strategy
Strategy (from Greek στρατηγία ''stratēgia'', "art of troop leader; office of general, command, generalship") is a general plan to achieve one or more long-term or overall goals under conditions of uncertainty. In the sense of the "art ...
which maximizes the minimal
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
. This procedure leads to the MaxEmin – Principle and is an extension of the
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematici ...
.
;The MaxWmin Principle
: This principle leads to the maximal guaranteed
weight function
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is ...
, regarding the extreme weights.
;The Prognostic Decision Principle (PDP)
: This principle is based on the
prognosis
Prognosis (Greek: πρόγνωσις "fore-knowing, foreseeing") is a medical term for predicting the likely or expected development of a disease, including whether the signs and symptoms will improve or worsen (and how quickly) or remain stabl ...
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
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
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.
See also
*
Edward Kofler
*
Fuzzy set
In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set.
At the same time, defined a ...
*
Fuzzy logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
*
Game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
*
Defuzzification
Defuzzification is the process of producing a quantifiable result in crisp logic, given fuzzy sets and corresponding membership degrees. It is the process that maps a fuzzy set to a crisp set. It is typically needed in fuzzy control systems. Th ...
*
Stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
*
Deterministic
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
*
Probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
*
Uncertainty
Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable or ...
*
Vagueness
In linguistics and philosophy, a vague predicate is one which gives rise to borderline cases. For example, the English adjective "tall" is vague since it is not clearly true or false for someone of middling height. By contrast, the word "prime" is ...
*
Optimization (mathematics)
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
*
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
*
List of set theory topics
This page is a list of articles related to set theory.
Articles on individual set theory topics
Lists related to set theory
* Glossary of set theory Talk:Glossary of set theory,
* List of large cardinal properties Talk:List of large cardinal ...
Selected references
* Edward Kofler – Equilibrium Points, Stability and Regulation in Fuzzy Optimisation Systems under Linear Partial Stochastic Information (LPI),
Proceedings
In academia and librarianship, conference proceedings is a collection of academic papers published in the context of an academic conference or workshop. Conference proceedings typically contain the contributions made by researchers at the confere ...
of the International Congress of
Cybernetics
Cybernetics is a wide-ranging field concerned with circular causality, such as feedback, in regulatory and purposive systems. Cybernetics is named after an example of circular causal feedback, that of steering a ship, where the helmsperson m ...
and Systems, AFCET, Paris 1984, pp. 233–240
* Edward Kofler –
Decision Making
In psychology, decision-making (also spelled decision making and decisionmaking) is regarded as the cognitive process resulting in the selection of a belief or a course of action among several possible alternative options. It could be either rati ...
under Linear Partial Information. Proceedings of the European Congress EUFIT, Aachen, 1994, pp. 891–896.
* Edward Kofler – Linear Partial Information with Applications. Proceedings of ISFL 1997 (International
Symposium
In ancient Greece, the symposium ( grc-gre, συμπόσιον ''symposion'' or ''symposio'', from συμπίνειν ''sympinein'', "to drink together") was a part of a banquet that took place after the meal, when drinking for pleasure was acc ...
on
Fuzzy Logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
), 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
External links
Tools for establishing dominance with linear partial information and attribute hierarchy{{Webarchive, url=https://web.archive.org/web/20110928131916/http://direct.bl.uk/bld/PlaceOrder.do?UIN=148552859&ETOC=RN&from=searchengine%2F , date=2011-09-28
Linear Partial Information with applications*
ttp://econpapers.repec.org/article/eeeintfor/v_3A4_3Ay_3A1988_3Ai_3A1_3Ap_3A15-32.htm/ Practical decision making with Linear Partial Information (LPI)br>
Stochastic programming with fuzzy linear partial information on probability distributionOne-shot decisions under Linear Partial Information
Information theory
Fuzzy logic
Randomized algorithms
Algorithmic information theory
Decision theory
Mathematical modeling