Linear partial information with applications

Abstract

The theory of linear partial information (LPI) belongs to the so-called soft modeling. In comparison with the fuzzy sets-theory the LPI-fuzziness is algorithmically more simple and especially in decision making more practically oriented. Instead of often dubious membership functions the decision maker linearizes any fuzziness by establishment of linear restrictions for fuzzy probability distributions or normalized weights (stochastic or non-stochastic LPI). In the decision aspect the axiomatic-based maximization of the minimal weighted sums (MaxWmin-principle), maximization of the minimal expected value (MaxEmin) and prognostic decision principle (PDP), the last by taking into account the risk readiness of the decision maker are applied. According to the linearity of the LPI-fuzziness only the extreme points of the corresponding LPI-convex polyhedrons are considered. The introduced notions of fuzzy equilibrium and stability are important for the diminution of classic mistake decisions. Depending on the decision principle the concepts of MaxEmin-, MaxWmin-, PDP-stability are analyzed. The ultra- and multistability under LPI-conditions are explained on some examples. In the area of multi-stage fuzzy decisions on the base of the Roll back procedure the adaptive stability with respect to learning and regulation aspects are investigated. The instability is a violation of the given stability interval interpreted and the removing of it is often connected with the profit-and-loss procedure. In the area of multiple objectives decision making under partial information the LPI-weighting method and stability problems are analyzed. Some practical applications are considered. In the domain of game theory the notions of equilibrium point and stability are extended into the LPI-fuzziness. This extension is important for practical applications. Some theorems are proved and their possible applications stated. For any fuzziness concept the corresponding fuzzy logic should be developed. In this paper the principles of the LPI-fuzzy logic are considered and their significance for the practice discussed. In particular, the classic principles of truth values, syllogism, modus ponens and others are extended for the LPI-case. In the paper such practical applications as forecasting of the economic growth, fuzzy linear programming, adaptive stability in the fuzzy multi-stage planning, investment decisions and fuzzy multiple-objective decisions are considered. In the forthcoming paper applications in socioeconomic decisions are analyzed.