Polynomial Regression of Expert Estimates of Complex Quality

Abstract. The multicriteria ranking problem of objects with several useful qualities is considered. Relating to the field of multicriteria optimization, this problem also arises when management decisions are chosen among several alternatives. The goal of this study is to develop a solution method based on calculating complex (generalized mean) quality indicators that represent polynomials from the class of normalized mean functions. The latter belong to strictly monotonic, shift-invariant aggregation operators. Such polynomials are called SPs for short. For example, the weighted arithmetic mean indicators of complex quality are SPs of degree 1. Apparently, SPs have all the properties of such linear functions that are essential for multicriteria ranking. Within the method presented, called the interactive approximation of expert estimates, we SPs of arbitrary degree for calculating complex quality indicators. This approach is similar to the expert-statistical method for determining weights. It provides the best root-mean-square approximation of any number of expert estimates, reducing their uncertainty and increasing their mutual consistency during the expertise procedure. The SPs of degrees 1, 2, and 3 are described below. The interactive approximation method of expert estimates is tested for SPs of degree 2 in the problem of calculating a complex quality indicator for smartphones ranked by seven partial qualities.

Keywords: multicriteria optimization, decision-making, normalized mean function, shift-invariant polynomial, aggregation operator, weight coefficient, complex indicator, expert estimate.

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Zot’ev, D.B. and Makhin, A.A., Polynomial Regression of Expert Estimates of Complex Quality. Control Sciences 1, 13–24 (2025).