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304 M. Mahdiloo et al.Yang, H. and Pollitt, M. (2009) ‘Incorporating both undesirable outputs and uncontrollablevariables into DEA: the performance of Chinese coal-fired power plants’, European Journalof Operational Research, Vol. 197, No. 3, pp.1095–1105.Zhu, J. and Cook, W.D. (2007) Modeling Data Irregularities and Structural Complexities in DataEnvelopment Analysis, A Problem-Solving Handbook, Springer, New York, USA.NomenclatureParameterj = 1, …, nnr = 1, …, si = 1, …, mDMU oxi oyr oDefinitionThe collection of suppliers (DMUs)The set of suppliers (DMUs)The set of outputsThe set of inputsThe DMU under investigationThe ith input of the DMU oThe rth output of the DMU oν Iu rx ijy rjDroThe weight for the ith inputThe weight for the rth outputThe ith input of the jth DMUThe rth output of the jth DMUy The rth desirable output of the DMU oyUroThe rth undesirable output of the DMU oDu The weight for rth desirable outputrUurThe weight for rth undesirable outputDy The rth desirable output of jth DMUrjyUrjThe rth undesirable output of jth DMUw os −iUs +rs +rDs +rë = [λ j ]UDThe variable which determines the ‘return to scale’ of the DMU oThe excesses in the ith inputThe excesses in the rth undesirable outputThe shortages in the rth outputThe shortages in the rth desirable outputThe vector of the DMU loadings, determining the ‘best practice’ for DMU oThe set of undesirable outputsThe set of desirable outputs
A novel DEA model for solving supplier selection problems 305Appendix 1: The translation invariance conceptFigure 1 depicts the lack of the translation invariance property of the input oriented BCCmodel with respect to inputs. In the input-oriented BCC efficiency of, which is thedistance of DMU D from the efficiency frontier constructed by efficient units A and B.This ratio is not values by deducting unity from them. Now, the efficiency frontieroriented BCC efficiency of DMUD ′ , DMU D after translation becomes, which is thedistance of from the efficiency frontier constructed by efficient units A′ and B′. SinceOR/OD # OR′/OD′, the input-oriented BCC model is not translation invariant with respectto the inputs.Figure 1Translation in the BCC model
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