440 ISSN 1064–5624, Doklady Mathematics, 2009, Vol. 79, No. 3, pp. 440–444. © Pleiades Publishing, Ltd., 2009. Published in Russian in Doklady Akademii Nauk, 2009, Vol. 426, No. 5, pp. 608–612. INTRODUCTION AND PROBLEM STATEMENT At present many practitioners and experts in busi- ness and economics use the ratio analysis for efﬁciency analysis of production units. In this approach, a number of ratios k i = , i = 1, 2, …, l are calculated for a spe- ciﬁc unit. Then an arithmetical function f (k 1 , k 2 , …, k l ) from these ratios is determined. It is assumed that this function is continuous and monotonous. This function is often called a performance index or just a rating func- tion. This very approach is used to analyze the activity of regions by Ministry of economic development and trade, of production units by Altmann [6], the stability and sol- vency of production units [7], the reliability of banks in Kromonov’s method [8, 9], the behavior of banks in their context in CAMEL method [8, 9] and so on. The Data envelopment analysis (DEA) approach generalizes the ratio analysis to the multidimensional case when the activity of production units is described by sets of inputs (x 1 , x 2 , …, x m ) and outputs (y 1 , y 2 , …, y r ). The DEA approach turned out to be very constructive under solution of many problems in practical and theo- retical economics. In particular, a multidimensional y i x i --- production possibility set can be constructed using a number of production units and one can investigate pro- duction units behavior in this set [1–3]. Now, the DEA approach has become a widely used methodology for a thorough analysis of the behavior of production units in their context. Charnes and Cooper [4, 5] were founders of this approach. At present there are several thousands of publications on this approach in international scientiﬁc journals. In this paper we compare the ratio analysis and the DEA approach, we also indicate how to construct rating functions using DEA models. MAIN RESULTS The DEA approach considers a set of n observations of actual production units X j = (x 1j , x 2j , …, x mj ) ≥ 0, j = 1, 2, …, n, where the vector of outputs Y j = (y 1j , y 2j , …, y rj ) ≥ 0 is produced from the vector of inputs X j = (x 1j , x 2j , …, x mj ) ≥ 0, j = 1, 2, …, n. The production possibil- ity set T is the set {(X, Y)| the outputs Y ≥ 0 can be pro- duced from the inputs X ≥ 0}. Set T is empirically speciﬁed by the following pos- tulates. Postulate 1 (convexity). If (X, Y) ∈ T and (X ', Y ') ∈ T , then (λX + (1 – λ)X ', λY + (1 – λ)Y ') ∈ T for all λ ∈ [0, 1]. Postulate 2 (monotonicity). If (X, Y) ∈ T and X ' ≥ X, Y ' ≤ Y , then (X ' Y ') ∈ T . Postulate 3 (minimal extrapolation). T is the inter- section set of all T ', satisfying Postulates 1 and 2, and subject to the condition that each of the observed vec- tors (X j , Y j ) ∈ T ', j = 1, 2, …, n. Construction of Rating Function Using DEA Models ¶ V. E. Krivonozhko, A. A. Piskunov, and A. V. Lychev Presented by Academician S.K. Korovin February 1, 2009 Received February 10, 2009 Abstract—The paper shows that many simple rating functions constructed on the basis of the ratio analysis distort signiﬁcantly the evaluation of the production units behavior. Opposite to the ratio analysis, the DEA approach investigate production units behaviour in the multidimentional space of indicators. In this paper, we compare the ration analysis and the DEA approach using properties of the mappings of the multidimentional sets, we also show that it is purposeful to construct rating functions on the basis of efﬁciency scores computed in DEA models. DOI: 10.1134/S1064562409030387 Institute of System Analysis, Russian Academy of Sciences, pr. 60-letiya Oktyabrya 9, Moscow, 117311 Russia; e-mail: KrivonozhkoVE@mail.ru, Lytchev@mail.ru COMPUTER SCIENCE ¶ The article was translated by the authors.