Abstract
The inverse DEA (Data Envelopment Analysis) method is primarily used to analyse the changing relationship between the inputs and outputs of a DMU (DecisionMaking Unit) when its efficiency is kept constant or set to a target value. However, the existing inverse DEA method cannot be applied directly to estimate all the changing relationships. For example, the existing DEA models fail to estimate the input variations when the supervisor wants to maintain the DMU’s outputoriented efficiency during the downscaling of production. This paper analyses all the possible changing relationships that need to be solved by the inverse DEA method and develops different models for both the output and input orientations, accomplishing the extension and integration of the inverse DEA model. For illustration of our results, a numerical example is given.
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Notes
 1.
This lemma has been discussed in Dulá and Thrall (2001). In this paper, we present and prove this lemma in a completely new way, which is more close to our topic and is the foundation for proving Theorems 1 and 2.
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Supported by Key Laboratory of Management, Decision and Information Systems, Chinese Academy of Science.
APPENDIX
APPENDIX
Technical proofs
Appendix A.1. Proof of Lemma 1
Proof

According to the conventional DEA theory, the calculation of the efficiency score of DMU′ _{o} follows the following model:
Suppose that the optimal solution of model (DLP _{0}) is (φ*, Λ*, v*) and that of (DLP′_{0}) is , then (φ*, Λ*, λ _{n+1}=0, v*) is a feasible solution of (DLP′_{0}), which indicates that . Considering constraint (2), we know that φ=1 is a feasible solution of (DLP_{0}). Thus, we have
Considering model (DLP′_{0}), we have λ _{n+1}⩽1 according to constraint (11). Otherwise, we would have ΛX ^{T}<0, which contradicts the constraint Λ⩾0. Next, We make a discussion in terms of .
If , then we have ΛX ^{T}=0 and Λ=0. Considering constraint (12), holds. Therefore, .
If , then make the following transformation:
We can conclude that is a feasible solution of model (DLP _{0}), so holds.
If , then formula (13) indicates that .
If , then we have
and , which contradicts formula (13). Therefore, is impossible.
If , then is a feasible solution of model (DLP _{0}). Thus, we find that . Therefore, holds due to formula (13).
To summarise, model (DLP_{0}) has the same optimal solution as model (DLP′_{0}). Therefore, when DMU′ _{o} satisfies constraint (2), models (DLP _{0}) and (DLP′_{0}) are equivalent, and both can be used to obtain the efficiency score of DMU′ _{o}. □
Appendix A.2. Proof of Corollary 1
Proof

We only need to prove that condition (1) always holds when δ _{2}=1 and δ _{3}=0.
Assuming that the Pareto solution S*=(ΔX*_{o}, Λ*, v*) does not meet condition (1), then there exists Λ* that satisfies
We can conclude that there exists a value τ(0<τ<1), from which we can obtain and satisfying and . The following inequality holds regarding :
Thus, there exists , and is a feasible solution of model (Output _{Y}), which contradicts the premise that ΔX* _{o} is the Pareto value. □
Appendix A.3. Proof of Corollary 2
Proof

For the VRS and NDRS efficiency scores of model (Output _{Y}), considering the arbitrary feasible solution , because , we have and
where Y _{r}=(y _{r1}, y _{r2}, …, y _{rn})^{T} is an ndimensional column vector. Thus, the Pareto solution does not meet condition (1). □
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Zhang, M., Cui, Jc. The extension and integration of the inverse DEA method. J Oper Res Soc 67, 1212–1220 (2016). https://doi.org/10.1057/jors.2016.2
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Keywords
 data envelopment analysis
 inverse DEA
 efficiency
 production possibility set
 multiobjective