BAM: a bounded adjusted measure of efficiency for use with bounded additive models William W. Cooper • Jesu ´s T. Pastor • Fernando Borras • Juan Aparicio • Diego Pastor Published online: 29 August 2010 Ó Springer Science+Business Media, LLC 2010 Abstract A decade ago the Range Adjusted Measure (RAM) was introduced for use with Additive Models. The empirical experience gained since then recommends developing a new measure with similar characteristics but with more discriminatory power. This task is accomplished in this paper by introducing the Bounded Adjusted Measure (BAM) in connection with a new family of Data Envel- opment Analysis (DEA) additive models that incorporate lower bounds for inputs and upper bounds for outputs while accepting any returns to scale imposed on the production technology. Keywords DEA  Additive models  Efficiency measures  Returns to scale  Bounded additive models JEL Classification C51  C61 1 Introduction The RAM measure, defined a decade ago by Cooper et al. (1999) was an attempt to define a generalized efficiency measure in connection with an additive model (Charnes et al. (1985), i.e., a measure that accounts for all the inefficiencies detected by the slacks of the model. As is well known, the classic additive model evaluates the pro- jection by maximizing the L 1 -distance to the strong effi- cient frontier and thereby simultaneously maximizes outputs and minimizes inputs. Moreover, the classic addi- tive model assumes variable returns to scale (VRS), an assumption that can easily be modified when formulated as a linear program [see, e.g., Ali and Seiford (1993)]. It is worth mentioning that the directional distance function measure of inefficiency, introduced by Chambers et al. (1998), does not account for all types of inefficiencies that the model can identify (see the example in Ray (2004), p. 95), but allows also for simultaneous output expansion and input minimization. As far as we know, during the last decade, few, if any, attempts have been made to develop a generalized effi- ciency measure in direct connection with additive linear efficiency models. Nonetheless, there have been three attempts to define generalized efficiency measures in connection to nonlinear efficiency models. The most popular one resorts to a fractional linear programming model that can be linearized, as shown by Pastor et al. (1999) and by Tone (2001) independently. The corre- sponding measure is now known as SBM (Slack Based Measure), although its first baptism was as ERG (Enhanced Russell Graph Measure). The second one, the GDF (Geometric Distance Function), was published by Silva-Portela and Thanassoulis (2006), who proposed it in an unpublished research paper in 2002. The GDF relies exclusively on the unit being rated and its projection, and can be used in connection with any DEA model. The last mentioned authors propose using GDF in connection to a nonlinear efficiency model with independent radial input reduction and radial output expansion. This model does W. W. Cooper Red McCombs School of Business, The University of Texas at Austin, Austin, TX 78712-1174, USA J. T. Pastor (&)  F. Borras  J. Aparicio Center of Operations Research (CIO), Universidad Miguel Hernandez de Elche, 03202 Elche, Alicante, Spain e-mail: jtpastor@umh.es D. Pastor Divisio ´n de Educacio ´n Fı ´sica y Deportiva, Universidad Miguel Hernandez de Elche, 03202 Elche, Alicante, Spain 123 J Prod Anal (2011) 35:85–94 DOI 10.1007/s11123-010-0190-2