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

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

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

. 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.

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 ...

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