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An aggressive game cross-efficiency evaluation in data envelopment analysis

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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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References

  • Andersen, P., & Petersen, N. C. (1993). A procedure for ranking efficient units in data envelopment analysis. Management Science, 39(10), 1261–1264.

    Article  Google Scholar 

  • Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring efficiency of decision making units. European Journal of Operational Research, 2(4), 429–444.

    Article  Google Scholar 

  • Chen, T. Y. (2002). An assessment of technical efficiency and cross-efficiency in Taiwan’s electricity distribution sector. European Journal of Operational Research, 137(2), 421–433.

    Article  Google Scholar 

  • Cooper, W. W., Thompson, R. G., & Thrall, R. M. (1996). Extensions and new development in DEA. Annals of Operations Research, 66(1), 3–45.

    Google Scholar 

  • Contreras, I. (2012). Optimizing the rank position of the DMU as secondary goal in DEA cross-evaluation. Applied Mathematical Modelling, 36(6), 2642–2648.

    Article  Google Scholar 

  • Despotis, D. K. (2002). Improving the discriminating power of DEA: Focus on globally efficient units. Journal of Operational Research Society, 53(3), 314–325.

    Article  Google Scholar 

  • Doyle, J. R., & Green, R. H. (1994). Efficiency and cross-efficiency in DEA: Derivations, meaning and the uses. The Journal of the Operational Research Society, 45(5), 567–578.

    Article  Google Scholar 

  • Doyle, J. R., & Green, R. H. (1995a). Cross-evaluation in DEA: Improving discrimination among DMUs. INFOR, 35(3), 205–222.

    Google Scholar 

  • Doyle, J. R., & Green, R. H. (1995b). Upper and lower bound evaluation of multiattibute objects: Comparison models using linear programming. Organizational Behavior and Human Decision Processes, 64(3), 61–273.

    Article  Google Scholar 

  • Dyson, R. G., & Thanassoulis, E. (1988). Reducing weight flexibility in data envelopment analysis. Journal of the Operational Research Society, 39(6), 563–576.

    Article  Google Scholar 

  • Emrouznejad, A. (2014). Advances in data envelopment analysis. Annals of Operations Research, 214(1), 1–4.

    Article  Google Scholar 

  • Entani, T., Maeda, Y., & Tanaka, H. (2002). Dual models of interval DEA and its extension to interval data. European Journal of Operation Research, 136(1), 32–45.

    Article  Google Scholar 

  • Green, R. H., Doyle, J. R., & Cook, W. D. (1996). Preference voting and project ranking using DEA and cross-evaluation. European Journal of Operational Research, 90(3), 461–472.

    Article  Google Scholar 

  • Jahanshahloo, G. R., Lofti, F. H., Yafari, Y., & Maddahi, R. (2011). Selecting symmetric weights as a secondary goal in DEA cross-efficiency evaluation. Applied Mathematical Modelling, 35(1), 544–549.

    Article  Google Scholar 

  • Jahanshahloo, G. R., Junior, H. V., Lotfi, F. H., & Akbarian, D. (2007). A new DEA ranking system based on changing the reference set. European Journal of Operational Research, 181(1), 331–337.

    Article  Google Scholar 

  • Kao, C., & Hung, H. T. (2005). Data envelopment analysis with common weights: The compromise solution approach. Journal of Operational Research Society, 56(10), 1196–1203.

    Article  Google Scholar 

  • Lei, X., Li, Y., Xie, Q., & Liang, L. (2015). Measuring Olympics achievements based on a parallel DEA approach. Annals of Operations Research, 226(1), 379–396.

    Article  Google Scholar 

  • Liang, L., Wu, J., Cook, W. D., & Zhu, J. (2008a). Alternative secondary goals in DEA cross-efficiency evaluation. International Journal of Production Economics, 113(2), 1025–1030.

    Article  Google Scholar 

  • Liang, L., Wu, J., Cook, W. D., & Zhu, J. (2008b). The DEA game cross-efficiency model and its Nash equilibrium. Operations Research, 56(5), 1278–1288.

    Article  Google Scholar 

  • Lim, S. (2012). Minimax and maximin formulations of cross-efficiency in DEA. Computers and Industrial Engineering, 62(3), 726–731.

    Article  Google Scholar 

  • Lim, S., Oh, K. W., & Zhu, J. (2014). Use of DEA cross-efficiency evaluation in portfolio selection: An application to Korean stock market. European Journal of Operational Research, 236(1), 361–368.

    Article  Google Scholar 

  • Liu, F. F., & Peng, H. H. (2008). Ranking of units on the DEA frontier with common weights. Computers & Operations Research, 35(5), 1624–1637.

    Article  Google Scholar 

  • Lotfi, F. H., Jahanshahloo, G. R., & Zamani, P. (2011). A new ranking method based on cross-efficiency in data envelopment analysis. African Journal of Business Management, 5(19), 7923–7930.

    Google Scholar 

  • Macro, F., Fabio, S., Nicola, C., & Roberto, P. (2012). Using a DEA-cross efficiency approach in public procurement tenders. European Journal of Operational Research, 218(2), 523–529.

    Article  Google Scholar 

  • Oral, M., Kettani, O., & Lang, P. O. (1991). A methodology for collective evaluation and selection of industrial R&D projects. Management Science, 37(7), 871–885.

    Article  Google Scholar 

  • Rakhshan, S. A., Kamyad, A. V., & Effati, S. (2015). Ranking decision-making units by using combination of analytical hierarchical process method and Tchebycheff model in data envelopment analysis. Annals of Operations Research, 226(1), 505–525.

    Article  Google Scholar 

  • Ramón, N., Ruiz, J. L., & Sirvent, I. (2010). On the choice of weights profiles in cross-efficiency evaluations. European Journal of Operational Research, 207(3), 1564–1572.

    Article  Google Scholar 

  • Ruiz, J. L. (2013). Cross-efficiency evaluation with directional distance functions. European Journal of Operational Research, 228(1), 181–189.

    Article  Google Scholar 

  • Sexton, T. S., Silkman, R. H., & Hogan, A. J. (1986). Data envelopment analysis: Critique and extensions. New Directions for Program Evaluation, 1986(32), 73–105.

    Article  Google Scholar 

  • Shang, J., & Sueyoshi, T. (1995). A unified framework for the selection of flexible manufacturing system. European Journal of Operational Research, 85(2), 297–315.

    Article  Google Scholar 

  • Sinuany-Stern, Z., & Friedman, L. (1998). DEA and the discriminant analysis of ratios for ranking units. European Journal of Operational Research, 111(3), 470–478.

    Article  Google Scholar 

  • Sinuany-Stern, Z., Mehrez, A., & Hadad, Y. (2000). An AHP/DEA methodology for ranking decision-making units. International Transactions in Operational Research, 7(2), 109–124.

    Article  Google Scholar 

  • Tan, Y., Zhang, Y., & Khodaverdi, R. (2017). Service performance evaluation using data envelopment analysis and balance scorecard approach: An application to automotive industry. Annals of Operations Research, 248(1), 449–470.

    Article  Google Scholar 

  • Wang, Y. M., & Chin, K. S. (2010a). Some alternative models for DEA cross-efficiency evaluation. International Journal of Production Research, 128(1), 332–338.

    Article  Google Scholar 

  • Wang, Y. M., & Chin, K. S. (2010b). A neutral DEA model for cross-efficiency evaluation and its extension. Expert Systems with Applications, 37(5), 3666–3675.

    Article  Google Scholar 

  • Wang, Y. M., Chin, K. S., & Jiang, P. (2011). Weight determination in the cross-efficiency evaluation. Computers & Industrial Engineering, 61(3), 497–502.

    Article  Google Scholar 

  • Wang, Y. M., Luo, Y., & Liang, L. (2009). Ranking decision making units by imposing a minimum weight restriction in the data envelopment analysis. Journal of Computational and Applied Mathematics, 223(1), 469–484.

    Article  Google Scholar 

  • Wang, Y. M., & Yang, J. B. (2007). Measuring the performances of decision-making units using interval efficiencies. Journal of Computational and Applied Mathematics, 198(1), 253–267.

    Article  Google Scholar 

  • Wong, Y. H. B., & Beasley, J. E. (1990). Restricting weight flexibility in data envelopment analysis. Journal of the Operational Research Society, 41(9), 829–835.

    Article  Google Scholar 

  • Wu, J., Chu, J., Zhu, Q., Li, Y., & Liang, L. (2016a). Determining common weights in data envelopment analysis based on the satisfaction degree. The Journal of the Operational Research Society, 67(12), 1446–1458.

    Article  Google Scholar 

  • Wu, J., Chu, J., Zhu, Q., Yin, P., & Liang, L. (2016b). DEA cross-efficiency evaluation based on satisfaction degree: An application to technology selection. International Journal of Production Research, 54(20), 5990–6007.

    Article  Google Scholar 

  • Wu, J., Liang, L., & Chen, Y. (2009a). DEA game cross-efficiency approach to Olympic rankings. Omega: The International Journal of Management Science, 37(4), 909–918.

    Article  Google Scholar 

  • Wu, J., Liang, L., Wu, D. X., & Yang, F. (2008). Olympics ranking and benchmarking based on cross-efficiency evaluation method and cluster analysis: the case of Sydney 2000. International Journal of Enterprise Network Management, 2(4), 377–392.

    Article  Google Scholar 

  • Wu, J., Liang, L., Zha, Y., & Yang, F. (2009b). Determination of cross-efficiency under the principle of rank priority in cross-evaluation. Expert Systems with Applications, 36(3), 4826–4829.

    Article  Google Scholar 

  • Wu, J., Sun, J. S., & Liang, L. (2012). Cross-efficiency evaluation method based on weight-balanced data envelopment analysis model. Computers & Industrial Engineering, 63(2), 513–519.

    Article  Google Scholar 

Download references

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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Correspondence to Ying-Ming Wang.

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

$$\begin{aligned} {{E}}_{dj}^{\min }= & {} \mathop {\min }\limits _{{{\varvec{u}}},{{\varvec{v}}}} \;{{\varvec{u}}}^{T}{{\varvec{y}}}_j\nonumber \\ \hbox {s.t.}&{{\varvec{v}}}^{T}{{\varvec{x}}}_j =1,\nonumber \\&{{E}}_d^*\cdot {{\varvec{v}}}^{T}{{\varvec{x}}}_d -{{\varvec{u}}}^{T}{{\varvec{y}}}_d \le 0,\nonumber \\&\mathop {\max }\limits _k \frac{{{\varvec{u}}}^{T}{{\varvec{y}}}_k }{{{\varvec{v}}}^{T}{{\varvec{x}}}_k }=1,\nonumber \\&{{\varvec{u}}}\ge \mathbf{0},{{\varvec{v}}}\ge \mathbf{0}. \end{aligned}$$

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

$$\begin{aligned} \theta _j^*=q{{E}}_{jj}^*+(1-q) \frac{\mathop {\sum }\limits _{d\ne j} {{E}_{dj}^*(\theta _d^*)} }{n-1}. \end{aligned}$$

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,

$$\begin{aligned} \alpha _j^2 =\frac{{E}_{jj}^*+\mathop {\sum }\limits _{d\ne j} {{E}_{dj}^*(\alpha _d^1 )} }{n}, \quad \bar{{{E}}}_j^{\min } =\frac{{E}_{jj}^*+\mathop {\sum }\limits _{d\ne j} {{E}_{dj}^{\min } } }{n}. \end{aligned}$$

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,

$$\begin{aligned} \alpha _j^{k+1} =\frac{{E}_{jj}^*+\mathop {\sum }\limits _{d\ne j} {{E} _{dj}^*(\alpha _d^k )} }{n}, \quad \alpha _{j}^{k} =\frac{{E}_{jj}^*+\mathop {\sum }\limits _{d\ne j} {{E}_{dj}^*(\alpha _d^{k-1} )} }{n}. \end{aligned}$$

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

$$\begin{aligned} \alpha _j^{k+1} =\frac{{E}_{jj}^*+\mathop {\sum }\limits _{d\ne j} {{E}_{dj}^*(\alpha _d^k )} }{n}, \quad \underline{\alpha }_j^{k+1} =\frac{{E}_{jj}^*+\mathop {\sum }\limits _{d\ne j} {{E}_{dj}^*(\underline{\alpha }_d^k )} }{n}, \quad \alpha _j^{k+2} =\frac{{E}_{jj}^*+\mathop {\sum }\limits _{d\ne j} {E_{dj}^*(\alpha _d^{k+1} )} }{n}. \end{aligned}$$

Based upon the supposition and Lemma 1, we have that for all j,

$$\begin{aligned} {{E}}_{dj}^*(\alpha _{j}^{k} )\ge {{E}}_{dj}^*(\underline{\alpha }_{j}^{k})\ge {{E}}_{dj}^*(\alpha _j^{k+1} ),k\ge 1. \end{aligned}$$

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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