Journal of Mathematical Sciences, Vol. 216, No. 5, August, 2016 APPROXIMATION OF SERIES OF EXPERT PREFERENCES BY DYNAMICAL FUZZY NUMBERS M. V. Koroteev, P. V. Terelyanskii, and V. A. Ivanyuk UDC 51-77 Abstract. In this paper, we consider a method of formalization of time series of arbitrary expert assessments under uncertainty conditions using both approximation and fuzzy arithmetic. We present basic definitions and a numerical example of calculation of fuzzy measures and propose comparative analysis of advantages and disadvantages of this method. In economic analysis, the problem of assessing time-varying parameters often appears. Traditionally, the approximation of time-series data by some smooth function of time is used. In practical problems, expert assessments often involve fuzzy information related to the uncertainty of the expert and/or varying measure of confidence in assessments made by the expert. In the uncertainty conditions, methods of fuzzy logic are quite adequate (see [2–7]). However, the combination of uncertainty and the complexity of the system described often makes the simultaneous formalized account of them in mathematical models of dynamic processes impossible. 1. Statement of problem. Assume that we have a parameter taking its values on a segment of the real line. Without loss of generality, we can assume that X ∈ [0; 1]; otherwise, we can perform a linear or nonlinear normalizing substitution. It is known that the value of this parameter can vary in time t; moreover, the (present) value of X at t = 0 is reliable, whereas its values at moments t> 0 are the subject of expert assessment, and the measure of uncertainty (distrust) of any assessment at a future time is a monotonically nondecreasing function as t increases. We have a set v of expert assessments of the values of X at moments t> 0, each of which consists of a time label and a assessment itself. There are the following types of assessments: (1) a point assessment: the most probable value of the parameter X at a given moment of time, t  = t ⇒ x ≈ x  ; (2) interval (two-point) assessment: at a given moment of time, the value of the parameter X is most probably lies within given limits, t  = t ⇒ x 1 ≤ x ≤ x 2 ; (3) three-point assessment: at a given moment of time, the value of the parameter X is most probably lies within given limits and there exists its most probable value within these limits, t  = t ⇒ x ≈ x 2 , x 1 ≤ x ≤ x 3 ; (4) lower bound: at a given moment of time, the parameter most probably takes values greater than a given level, t  = t ⇒ x ≥ x  ; (5) upper bound: at a given moment of time, the parameter most probably takes values less than a given level, t  = t, ⇒ x ≤ x  . We must construct an approximation of these assessments such that the following conditions hold: (1) at each moment of time, an assessment of the parameter is a fuzzy number defined on the domain of the parameter; (2) at each moment of time, there exists at least one most probable value of the parameter; Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applica- tions), Vol. 95, Models of Mathematical Economics, 2015. 692 1072–3374/16/2165–0692 c  2016 Springer Science+Business Media New York DOI 10.1007/s10958-016-2930-y