Abstract
Cross-efficiency evaluation is an effective method for ranking decision making units (DMUs) in data envelopment analysis, which is performed with peer-evaluation and self-evaluation. From different points of view, various cross-efficiency evaluations have been proposed with different secondary goals. Yet they usually lead to different average cross-efficiencies and different rankings. In this paper, we develop a concept of the aggressive game cross-efficiency, and propose an aggressive secondary model to minimize the cross-efficiencies of other DMUs under the constraints that the aggressive game cross-efficiency of the evaluated DMU is guaranteed. To achieve the aggressive game cross-efficiency, we develop an iterative algorithm. Mathematically, it is proved that all conventional average cross-efficiencies are sure to converge to the same aggressive game cross-efficiency by the iterative algorithm. Finally, numerical examples are presented to show the effectiveness of our approach in evaluating and ranking DMUs.
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Acknowledgements
The work described in this paper is supported by The National Natural Science Foundation of China (NSFC) under the Grant No. 71371053 and also supported by The Science and Technology Development Fund of Fuzhou University of China under the Grant No. 600915.
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Appendix
Appendix
Lemma 1
For all \(d,j=1,2,\ldots ,n\), we consider the optimal value of model (11), namely \({{E}}_{dj}^*(\alpha _d)\), as a function of \(\alpha _d \). Thus \({{E}}_{dj}^*(\alpha _d)\) is non-decreasing on \([0,{{E}}_d^*]\).
Proof
Assume that \(\alpha _d^{small} ,\alpha _d^{big} \in [0,E_d^*]\), and \(\alpha _d^{small} <\alpha _d^{big} \). Then the feasible region of model (11) with \(\alpha _d =\alpha _d^{small} \) embodies the feasible region with \(\alpha _d =\alpha _d^{big} \). Therefore, as the minimum of model (11) \({{E}}_{dj}^*(\alpha _d^{small} )\)is not larger than \({{E}}_{dj}^*(\alpha _d^{big} )\). That is to say, the function \({{E}}_{dj}^*(\alpha _d )\) is non-decreasing on \([0,{{E}}_d^*]\).
Proof of Theorem 1
If \(\,\alpha _d =0\), obviously, model (11) is equivalent to model (5). That means, \({{E}}_{dj}^*(0)=\theta _j^*\).
In the constraints of model (8), noticing that \({{E}}_d^*\) is the CCR efficiency of model (2), we have that model (8) is equivalent to
Then, it is easy to see that \({{E}}_{dj}^{\min } ({E}_d^*)={{E}}_{dj}^{\min }\).
Based on Lemma 1, \({{E}}_{dj}^*(\alpha _d )\) is non-decreasing on \([0,{{E}}_d^*]\). Given \(\alpha _d \in [0,{{E}}_d^*]\), then we have \({{E}}_{dj}^*(0)\le {{E}}_{dj}^*(\alpha _d )\le {{E}}_{dj}^*({{E}}_d^*)\), namely \(\theta _j^*\le {{E}}_{dj}^*(\alpha _d )\le {{E}}_{dj}^{\min }\).
Proof of Theorem 2
Based on model (5), for \(d=1,2,\ldots ,n\) and any weight vectors \({{\varvec{u}}}\ge \mathbf{0},{{\varvec{v}}}\ge \mathbf{0}\) such that \(\mathop {\hbox {max}}\limits _{k} \;{{\varvec{u}}}^{T}{{\varvec{y}}}_k /{{\varvec{v}}}^{T}{{\varvec{x}}}_k =1\), we have that \(\frac{{{\varvec{u}}}^{T}{{\varvec{y}}}_d }{{{\varvec{v}}}^{T}{{\varvec{x}}}_d }\ge \theta _d^*\), where \(\theta _d^*\) is the pessimistic efficiency of \(\hbox {DMU}_{d}\). If we set \(\alpha _d =\theta _d^*\), then the second constraint of model (11), namely \(\frac{{{\varvec{u}}}^{T}{{\varvec{y}}}_d }{{\varvec{v}}^{T}{{\varvec{x}}}_d }\ge \theta _d^*\), must be satisfied only if \(\mathop {\hbox {max}}\limits _{k} \;{{\varvec{u}}}^{T}{{\varvec{y}}}_k /{{\varvec{v}}}^{T}{{\varvec{x}}}_k =1\), and model (11) is equivalent to model (5). Hence, \({{E}}_{dj}^*(\theta _d^*)\)is just the pessimistic efficiency of \(\hbox {DMU}_{j}\), namely \({{E}}_{dj}^*(\theta _d^*)=\theta _j^*,d=1,2,\ldots ,n\).
If \(q=0\), then we have
Therefore, the pessimistic efficiency \(\theta _{j}^{*}\) is just the weighted aggressive game cross-efficiency of \(\hbox {DMU}_{j}\) with the parameter \(q=0\).
Proof of Theorem 3
First we will show \(\alpha _j^2 \le \alpha _j^1 \) for all j in \(\{1,2,\ldots ,n\}\). In fact,
Based on Theorem 1, we have \({{E}}_{dj}^*(\alpha _d^1 )\le {{E}}_{dj}^{\min }\). Therefore, \(\alpha _j^2 \le \bar{{{E}}}_j^{\min } \). Noting that \(\alpha _j^1 \in [\bar{{{E}}}_j^{\min } ,{{E}}_j^*]\), we obtain that \(\alpha _j^2 \le \bar{{{E}}}_j^{\min } \le \alpha _j^1 \) for all j.
Assume that \(\alpha _{j}^{k} \le \alpha _j^{k-1} ,j=1,2,\ldots ,n,k>1\). Thus we will deduce \(\alpha _j^{k+1} \le \alpha _{j}^{k} \) for all j. In fact,
Based on the assumption and Lemma 1, we have \(E_{dj}^{*} (\alpha _d^k )\le {{E}}_{dj}^*(\alpha _d^{k-1} ),j,d=1,2,\ldots ,n,j\ne d\). Thus \(\alpha _j^{k+1} \le \alpha _{j}^{k} ,j=1,2,\ldots ,n\). This means the sequence \(\{\alpha _j^1 ,\alpha _j^2 ,\ldots ,\alpha _{j}^{k} ,\ldots \}\)is non-increasing. Noting that \(\alpha _{j}^{k} \ge 0\), we have that the sequence \(\{\alpha _j^1 ,\alpha _j^2 ,\ldots ,\alpha _{j}^{k} ,\ldots \}\) converges.
Proof of Theorem 4
First, let \(k=1\) . Since \(\underline{\alpha }_j^1 =\bar{{{E}}}_j^{\min } ,\alpha _j^1 \ge {\alpha }_j^1 \ge \alpha _j^2 ,j=1,2,\ldots ,n\), as proved in Theorem 3.
Suppose that \(\alpha _{j}^{k} \ge {\alpha }_{j}^{k} \ge \alpha _j^{k+1} ,k\ge 1,j=1,2,\ldots ,n\). Then
Based upon the supposition and Lemma 1, we have that for all j,
Hence, for each \(j,\alpha _j^{k+1} \ge \underline{\alpha }_j^{k+1} \ge \alpha _j^{k+2} ,k\ge 1\), and Theorem 4 is obtained.
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Liu, W., Wang, YM. & Lv, S. An aggressive game cross-efficiency evaluation in data envelopment analysis. Ann Oper Res 259, 241–258 (2017). https://doi.org/10.1007/s10479-017-2524-1
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DOI: https://doi.org/10.1007/s10479-017-2524-1