A modification of well known Aggregated Indices Method (AIM) is developed for complex multi-attribute objects preference (quality) evaluation under deficiency of numerical information. The modification is based on so called ”double randomization” of weight coefficients, which are measuring the objects characteristics significance. The so modified AIM is named AIRM (Aggregated Indices Randomization Method). The AIRM may work with non-numeric (ordinal), and imprecise (interval) expert information. A case of Russian blue chips preference estimation under uncertainty demonstrates AIRMs applicability to investment portfolio formation.