Production, Manufacturing and Logistics The impossibility of convex constant returns-to-scale production technologies with exogenously fixed factors Victor V. Podinovski a,⇑ , Tatiana Bouzdine-Chameeva b,1 a Warwick Business School, University of Warwick, Coventry CV4 7AL, UK b BEM Bordeaux Management School, 680 cours de la Libération, Talence 33405, France article info Article history: Received 19 March 2010 Accepted 2 March 2011 Available online xxxx Keywords: Data envelopment analysis Exogenous factors Discretionary factors Returns to scale abstract The extensions to the variable (VRS) and the constant (CRS) returns-to-scale models developed by Banker and Morey are considered among the main approaches to the incorporation of exogenously fixed factors in models of data envelopment analysis (DEA). Recently, Syrjänen showed that the Banker and Morey CRS technology is not convex. Taking into account that its subset VRS technology is explicitly assumed con- vex, this observation leads to difficulties with explaining the fundamental production assumptions of the CRS extension. Motivated by the example of Syrjänen, the contribution of this paper is twofold. First, we show that the nonconvex Banker and Morey CRS technology is nevertheless a suitable reference technol- ogy for the assessment of scale efficiency. Second, we ask if a convex technology could be constructed that would ‘‘correct’’ the nonconvexity of the CRS technology of Banker and Morey. The answer to this is negative: one consequence of assuming both convexity and ray unboundness with fixed exogenous fac- tors is that we can always ‘‘mix-and-match’’ discretionary and nondiscretionary factors taken from dif- ferent units, arriving at totally unrealistic production plans. This demonstrates that generally there exists no meaningful convex CRS technology with exogenously fixed factors that can be used in its own right, apart from its use as a reference technology in the measurement of scale efficiency. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction In the original constant (CRS) and variable (VRS) returns- to-scale models of data envelopment analysis (DEA), the decision making units are assumed to be in control of their inputs and out- puts (Charnes et al., 1978; Banker et al., 1984). Banker and Morey (1986) suggested an extension to these models (below referred to as the BM models) suitable for the incorporation of exogenously fixed (uncontrollable) factors. The approach of Banker and Morey allows the controllable inputs and outputs to be scaled exactly as in the standard CRS model, but disallows the corresponding scaling of the uncontrollable factors. The latter is achieved by the incorpo- ration of special normalizing constraints that keep the sum of weighting coefficients equal to one for such factors. This develop- ment is widely considered as one of the main approaches to the treatment of exogenous inputs and outputs. Alternative models with exogenous factors were considered in Lovell (1994), Ruggiero (1996) and Cooper et al. (2000). In contrast with the BM model, the approaches of Lovell and Ruggiero do not assume that exogenous factors can be averaged (used in convex combinations). In Cooper et al. (2000), exogenous factors can be scaled in the same way as discretionary factors, while this is not al- lowed in the BM model. Further extensions and comparative anal- ysis of these and other models were presented in Golany and Roll (1993), Syrjänen (2004), Muñiz et al. (2006), Yang and Pollitt (2009) and Löber and Staat (2010). It is worth noting that Banker and Morey introduced their CRS model only as a reference technology, in the context of measuring the scale efficiency of units operating in the VRS technology. However, more recent developments by Golany and Roll (1993), Syrjänen (2004), Muñiz et al. (2006) and Löber and Staat (2010) suggest that there is a broader interest in the BM CRS model as a standalone model in its own right. If this is the case, then, as with any other production technology, the production assumptions on which the BM CRS model is based should be examined. Banker and Morey (1986) do not offer any axiomatic support for their development, but their CRS model appears to be intuitively straightforward. However, the axiomatic investigation of Syrjänen (2004) demonstrates that the BM CRS production technology is not convex. The nonconvexity of the CRS extension of the underlying convex VRS technology is clearly highly undesirable. In particular, it appears problematic that its VRS subset is explicitly assumed to be (and is) convex while the units in the CRS extension (outside the core VRS technology) are not subject to the same assumption. 0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.03.003 ⇑ Corresponding author. Tel.: +44 (0)24 7652 4281; fax: +44 (0)24 7652 4539. E-mail addresses: v.podinovski@warwick.ac.uk (V.V. Podinovski), Tatiana. chameeva@bem.edu (T. Bouzdine-Chameeva). 1 Tel.: +33 (0)55 6884 2235; fax: +33 (0)55 684 55 00. European Journal of Operational Research xxx (2011) xxx–xxx Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor Please cite this article in press as: Podinovski, V.V., Bouzdine-Chameeva, T. The impossibility of convex constant returns-to-scale production technologies with exogenously fixed factors. European Journal of Operational Research (2011), doi:10.1016/j.ejor.2011.03.003