[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content

Advertisement

Springer Nature Link
Log in

Efficiency evaluation with data uncertainty

  • Original Research
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

As one of the most popular techniques for performance evaluation, Data Envelopment Analysis (DEA) has been widely applied in many areas. However, the self-evaluation used in DEA leaves it open to much criticism. Moreover, most researchers have ignored the fact that reality abounds with uncertainty and have assumed that the data used for evaluation is deterministic and accurate. Both assumptions make it difficult to evaluate the efficiency of real-world production processes correctly and reasonably. In this paper, we propose a series of robust cross-efficiency (RCE) models based on robust optimization theory and cross-efficiency to deal with these problems. First of all, the proposed RCE models allow the conservatism level to be adjusted easily to suit the attitude of the decision-maker towards uncertainty. In addition, the RCE models have better discrimination power than the existing robust CCR models. We present two applications to demonstrate the effectiveness and stability of our models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. http://www.stats.gov.cn/tjsj/ndsj/2019/indexch.htm.

References

  • An, Qingxian, Chen, Haoxun, Wu, Jie, & Liang, Liang. (2015). Measuring slacks-based efficiency for commercial banks in China by using a two-stage DEA model with undesirable output. Annals of Operations Research, 235(1), 13–35.

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

    Article  Google Scholar 

  • Anderson, Timothy R., Hollingsworth, Keith, & Inman, Lane. (2002). The fixed weighting nature of a cross-evaluation model. Journal of Productivity Analysis, 17(3), 249–255.

    Article  Google Scholar 

  • Arabmaldar, Aliasghar, Jablonsky, Josef, & Hosseinzadeh Saljooghi, Faranak. (2017). A new robust dea model and super-efficiency measure. Optimization, 66(5), 723–736.

    Article  Google Scholar 

  • Banker, Rajiv D., Charnes, Abraham, & Cooper, William Wager. (1984). Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science, 30(9), 1078–1092.

    Article  Google Scholar 

  • Ben-Tal, Aharon, & Nemirovski, Arkadi. (2000). Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, 88(3), 411–424.

    Article  Google Scholar 

  • Bertsimas, Dimitris, & Sim, Melvyn. (2004). The price of robustness. Operations Research, 52(1), 35–53.

    Article  Google Scholar 

  • Charnes, Abraham, & Cooper, William W. (1962). Programming with linear fractional functionals. Naval Research Logistics Quarterly, 9(3–4), 181–186.

    Article  Google Scholar 

  • Charnes, Abraham, Cooper, William W., & Rhodes, Edwardo. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429–444.

    Article  Google Scholar 

  • Cooper, William W., Seiford, Lawrence M., & Tone, Kaoru. (2007). Data envelopment analysis: a comprehensive text with models, applications, references and DEA-solver software (Vol. 2). Berlin: Springer.

    Book  Google Scholar 

  • Dotoli, Mariagrazia, Epicoco, Nicola, & Falagario, Marco. (2017). A fuzzy technique for supply chain network design with quantity discounts. International Journal of Production Research, 55(7), 1862–1884.

    Article  Google Scholar 

  • Dotoli, Mariagrazia, Epicoco, Nicola, Falagario, Marco, & Sciancalepore, Fabio. (2015). A cross-efficiency fuzzy data envelopment analysis technique for performance evaluation of decision making units under uncertainty. Computers & Industrial Engineering, 79, 103–114.

    Article  Google Scholar 

  • Dotoli, Mariagrazia, Epicoco, Nicola, Falagario, Marco, & Sciancalepore, Fabio. (2016). A stochastic cross-efficiency data envelopment analysis approach for supplier selection under uncertainty. International Transactions in Operational Research, 23(4), 725–748.

    Article  Google Scholar 

  • Doyle, Jhon R., & Green, Rodney H. (1995). Cross-evaluation in dea: Improving discrimination among DMUs. INFOR: Information Systems and Operational Research, 33(3), 205–222.

    Google Scholar 

  • Hatami-Marbini, Adel, Emrouznejad, Ali, & Tavana, Madjid. (2011). A taxonomy and review of the fuzzy data envelopment analysis literature: two decades in the making. European Journal of Operational Research, 214(3), 457–472.

    Article  Google Scholar 

  • Kuntz, Ludwig, & Scholtes, Stefan. (2000). Measuring the robustness of empirical efficiency valuations. Management Science, 46(6), 807–823.

    Article  Google Scholar 

  • Liang, Liang, Wu, Jie, Cook, Wade D., & Zhu, Joe. (2008). The DEA game cross-efficiency model and its nash equilibrium. Operations Research, 56(5), 1278–1288.

  • Li, Feng, Han, Wu., Zhu, Qingyuan, Liang, Liang, & Kou, Gang. (2021). Data envelopment analysis cross efficiency evaluation with reciprocal behaviors. Annals of Operations Research, 302(1), 173–210.

    Article  Google Scholar 

  • Liu, Shiang-Tai., & Lee, Yueh-Chiang. (2021). Fuzzy measures for fuzzy cross efficiency in data envelopment analysis. Annals of Operations Research, 300(2), 369–398.

    Article  Google Scholar 

  • Lu, Chung-Cheng. (2015). Robust data envelopment analysis approaches for evaluating algorithmic performance. Computers & Industrial Engineering, 81, 78–89.

    Article  Google Scholar 

  • Mulvey, John M., Vanderbei, Robert J., & Zenios, Stavros A. (1995). Robust optimization of large-scale systems. Operations Research, 43(2), 264–281.

    Article  Google Scholar 

  • Nejad, Zahra Mohmmad, & Ghaffari-Hadigheh, Alireza. (2018). A novel DEA model based on uncertainty theory. Annals of Operations Research, 264(1), 367–389.

    Article  Google Scholar 

  • Olesen, Ole B., & Petersen, Niels Christian. (2016). Stochastic data envelopment analysis–a review. European Journal of Operational Research, 251(1), 2–21.

    Article  Google Scholar 

  • Omrani, H. (2013). Common weights data envelopment analysis with uncertain data: A robust optimization approach. Computers & Industrial Engineering, 66(4), 1163–1170.

    Article  Google Scholar 

  • Omrani, Hashem, Shafaat, Khatereh, & Alizadeh, Arash. (2019). Integrated data envelopment analysis and cooperative game for evaluating energy efficiency of transportation sector: a case of Iran. Annals of Operations Research, 274(1–2), 471–499.

    Article  Google Scholar 

  • Roll, Yaakov, Cook, Wade D., & Golany, Boaz. (1991). Controlling factor weights in data envelopment analysis. IIE Transactions, 23(1), 2–9.

    Article  Google Scholar 

  • Sadjadi, Seyed Jafar, Hossein Omrani, S., Abdollahzadeh, Mohammad Alinaghian, & Mohammadi, H. (2011). A robust super-efficiency data envelopment analysis model for ranking of provincial gas companies in Iran. Expert Systems with Applications, 38(9), 10875–10881.

    Article  Google Scholar 

  • Sadjadi, Seyed Jafar, & Omrani, Hashem. (2008). Data envelopment analysis with uncertain data: An application for Iranian electricity distribution companies. Energy Policy, 36(11), 4247–4254.

    Article  Google Scholar 

  • Sadjadi, Seyed Jafar, Omrani, Hossein, Makui, Ahmad, & Shahanaghi, Kamran. (2011). An interactive robust data envelopment analysis model for determining alternative targets in Iranian electricity distribution companies. Expert Systems with Applications, 38(8), 9830–9839.

    Article  Google Scholar 

  • Salahi, Maziar, Torabi, Narges, & Amiri, Akbar. (2016). An optimistic robust optimization approach to common set of weights in DEA. Measurement, 93, 67–73.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Shokouhi, Amir H., Hatami-Marbini, Adel, Tavana, Madjid, & Saati, Saber. (2010). A robust optimization approach for imprecise data envelopment analysis. Computers & Industrial Engineering, 59(3), 387–397.

    Article  Google Scholar 

  • Soyster, Allen L. (1973). Convex programming with set-inclusive constraints and applications to inexact linear programming. Operations Research, 21(5), 1154–1157.

    Article  Google Scholar 

  • Toloo, Mehdi, & Mensah, Emmanuel Kwasi. (2019). Robust optimization with nonnegative decision variables: a DEA approach. Computers & Industrial Engineering, 127, 313–325.

    Article  Google Scholar 

  • Wang, Ying-Ming., & Chin, Kwai-Sang. (2010). A neutral dea model for cross-efficiency evaluation and its extension. Expert Systems with Applications, 37(5), 3666–3675.

    Article  Google Scholar 

  • Wei, Guiwu, & Wang, Jiamin. (2017). A comparative study of robust efficiency analysis and data envelopment analysis with imprecise data. Expert Systems with Applications, 81, 28–38.

    Article  Google Scholar 

  • Wu, Jie, Zhang, Ganggang, Zhu, Qingyuan, & Zhou, Zhixiang. (2020). An efficiency analysis of higher education institutions in China from a regional perspective considering the external environmental impact. Scientometrics, 122(1), 57–70.

  • Zahedi-Seresht, Mazyar, Jahanshahloo, Gholam-Reza., & Jablonsky, Josef. (2017). A robust data envelopment analysis model with different scenarios. Applied Mathematical Modelling, 52, 306–319.

    Article  Google Scholar 

  • Zhang, Ganggang, Jie, Wu., & Zhu, Qingyuan. (2020). Performance evaluation and enrollment quota allocation for higher education institutions in China. Evaluation and Program Planning, 81, 101821.

    Article  Google Scholar 

  • Zhu, Qingyuan, Li, Xingchen, Li, Feng, Wu, Jie, & Zhou, Dequn. (2020). Energy and environmental efficiency of China’s transportation sectors under the constraints of energy consumption and environmental pollutions. Energy Economics,89, 104817.

Download references

Author information

Authors and Affiliations

  1. School of Management, University of Science and Technology of China, Hefei, 230026, Anhui Province, People’s Republic of China

    Jie Wu, Lulu Shen & Ganggang Zhang

  2. School of Economics, Hefei University of Technology, Hefei, 230601, Anhui Province, People’s Republic of China

    Zhixiang Zhou

  3. College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, 210000, Jiangsu Provincee, People’s Republic of China

    Qingyuan Zhu

Authors
  1. Jie Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lulu Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ganggang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhixiang Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Qingyuan Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ganggang Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A

1.1 Construction of model 11

We first formulate the uncertain CCR model as follows:

$$\begin{aligned} \begin{aligned} {\theta }_{d}=\text {Max}&\sum \limits _{r=1}^{s}{u_{r}^{d}{{{{\tilde{y}}}}_{rd}}} \\ \text {s.t.}&\sum \limits _{r=1}^{s}{u_{r}^{d}}{{{{\tilde{y}}}}_{rj}}-\sum \limits _{r=1}^{m}{v_{i}^{d}{{{{\tilde{x}}}}_{ij}}} \le 0;j=1\dots n \\&\sum \limits _{r=1}^{m}{v_{i}^{d}{{{{\tilde{x}}}}_{id}}} =1 \\&\quad u_{r}^{d}\ge 0,r=1\ldots s\\&\quad v_{i}^{d} \ge 0,i=1\ldots m. \end{aligned} \end{aligned}$$
(13)

The method developed by Bertsimas and Sim (2004) tried to find the optimal solutions when the worst situation happened. We first consider the constraints of model (13). Based on the predefined uncertainty set, we have the following formulations:

$$\begin{aligned} \underset{({\tilde{x}}_{ij},{\tilde{y}}_{rj})\in U}{{\text {Max}}}\left( \sum \limits _{r=1}^{s}{u_r^d{\tilde{y}}_{rj}} -\sum \limits _{i=1}^m{v_i^d{\tilde{x}}_{ij}}\right)&= \sum \limits _{r=1}^{s}{u_r^d{\bar{y}}_{rj}} +\underset{|\xi _{rj}|\le 1,\sum {\xi _{rj}}\le \Gamma _j^y}{{\text {Max}}}\underset{r\in {{R}_{j}}}{\sum } \,{{u}_{r}^{d}}{\xi _{rj}}{{{{\hat{y}}}}_{rj}}\\&\quad -\sum \limits _{r=1}^{s}{v_i^d{\bar{x}}_{ij}} +\underset{|\eta _{ij}|\le 1,\sum {\eta _{ij}}\le \Gamma _j^x}{{\text {Max}}}\underset{i\in {{I}_{j}}}{\sum } \,{{v}_{i}^{d}}{\eta _{ij}}{{{{\hat{x}}}}_{ij}} \end{aligned}$$

Using \(\beta _{j}^{y}\left( \Gamma _{j}^{y} \right) \) to denote \(\underset{|\xi _{rj}|\le 1\sum {\xi _{rj}}\le \Gamma _j^y}{\mathop {\text {Max}}} \underset{r\in {{R}_{j}}}{\mathop \sum }\,{{u}_{r}^{d}}{\xi _{rj}}{{{{\hat{y}}}}_{rj}} \), then we have

$$\begin{aligned} \begin{aligned} \beta _{j}^{y}\left( \Gamma _{j}^{y} \right) =\text {Max}&\underset{r\in {{R}_{j}}}{\sum }\,{{u}_{r}^{d}}{{\xi }_{rj}}{{{{\hat{y}}}}_{rj}} \\ \text {s.t.}&\underset{r\in {{R}_{j}}}{\sum }\,{{\xi }_{rj}}\le \Gamma _{j}^{y} \\&\quad {{\xi }_{rj}}\le 1,r\in {{R}_{j}} \\&\quad {{\xi }_{rj}}\ge 0,r\in {{R}_{j}}. \end{aligned} \end{aligned}$$
(14)

The dual model of (14) is presented as (15):

$$\begin{aligned} \begin{aligned} \beta _{j}^{y}\left( \Gamma _{j}^{y} \right) =\text {Min}&z_{j}^{y}\Gamma _{j}^{y}+\sum \limits _{r\in {{R}_{j}}}{{{q}_{rj}}} \\ \text {s.t.}&z_{j}^{y}+{{q}_{rj}}\ge u_{r}^{d}{{{{\hat{y}}}}_{rj}},\ r\in {{R}_{j}} \\&\quad z_{j}^{y}\ge 0 \\&\quad {{q}_{rj}}\ge 0,\ r\in {{R}_{j}}. \end{aligned} \end{aligned}$$
(15)

From duality theory, we know the optimal values of (14) and (15) are equal. For the same reason, we have (16):

$$\begin{aligned} \begin{aligned} \beta _{j}^{x}\left( \Gamma _{j}^{x} \right) =\text {Min}&z_{j}^{x}\Gamma _{j}^{x}+\sum \limits _{i\in {{I}_{j}}}{{{p}_{ij}}} \\ \text {s.t.}&z_{j}^{x}+{{p}_{ij}}\ge v_{i}^{d}{{{{\tilde{x}}}}_{ij}},\ i\in {{I}_{j}} \\&\quad z_{j}^{x}\ge 0 \\&\quad {{p}_{ij}}\ge 0,\ i\in {{I}_{j}}. \end{aligned} \end{aligned}$$
(16)

As mentioned before, it is required that the robust efficiency should be smaller than the nominal efficiency, so the first constraint of (13) can be rewritten as \(\sum \nolimits _{r=1}^{s}{u_r^d{\bar{y}}_{rj}}-\theta _j^{CCR}\sum \nolimits _{r=1}^{s}{v_i^d{\bar{x}}_{ij}}+\beta _{j}^{x}\left( \Gamma _{j}^{x} \right) +\beta _{j}^{y}\left( \Gamma _{j}^{y} \right) \le 0\). The robust formulation of the goal function of model (12) can be rewritten as \(\underset{u_r^d}{\mathop {\text {Max}}} \underset{{\tilde{y}}_{rd}\in U_y}{{\text {Min}}}{\sum \nolimits _{r=1}^s{{u}_{r}^{d}{\tilde{y}}_{rd}}}=\underset{u_r^d}{{\text {Max}}}{\sum \nolimits _{r=1}^s{{u}_{r}^{d}{\bar{y}}_{rd}}}-\beta _{d}^{y}\left( \Gamma _{d}^{y} \right) \). We integrate the constraints of (15) and (16) into model (13) to complete the construction process. \(\square \)

1.2 Construction of model 12

Following the proof of proposition 1, the robust formulation of the benevolent model’s goal function is

$$\begin{aligned} \underset{u_r^d}{{\text {Max}}} \underset{{\tilde{y}}_{rj}\in U_y}{{\text {Min}}}{\sum \limits _{r=1}^s{{u}_{r}^{d}\sum \limits _{j\ne d}{{\tilde{y}}_{rj}}}}=\underset{u_r^d}{{\text {Max}}}{\sum \limits _{r=1}^s{{u}_{r}^{d}\sum \limits _{j\ne d}{{\bar{y}}_{rj}}}}-\sum \limits _{j\ne d}{\beta _{j}^{y}\left( \Gamma _{j}^{y} \right) } \end{aligned}$$

The robust counterpart of the first constraint of model (5) (i.e., benevolent/aggressive model) can be formulated as the same as model (13). The robust counterpart of second constraint of model (5) is developed as \({\sum \nolimits _{i=1}^m{{v}_i^d({\sum \nolimits _{j\ne d}{\tilde{x}}_{ij}})}}={\sum \nolimits _{i=1}^{m}{{v}_{i}^{d}(\sum \nolimits _{j\ne d}{{\bar{x}}_{ij}}})}+\sum \nolimits _{j\ne d}{\beta _{j}^{y}(\Gamma _{j}^{y})}=1\). The third constraint of model (5) aims to keep the efficiency of the DMU under evaluation unchanged, thus it can be formulated as \({\sum \nolimits _{r=1}^{s}{u_r^d{\bar{y}}_{rd}}}-E_{dd}^R\sum \nolimits _{i=1}^m{v_i^d{\bar{x}}_{id}}=0\). As the aggressive model tries to minimize the other DMUs’ cross-efficiencies, the robust counterpart (i.e., when the worst case happens) can be formulated as \(\underset{u_r^d}{{\text {Min}}} \underset{{\tilde{y}}_{rj}\in U_y}{{\text {Max}}}{\sum \nolimits _{r=1}^s{{u}_{r}^{d}\sum \nolimits _{j\ne d}{{\tilde{y}}_{rj}}}} =\underset{u_r^d}{{\text {Min}}} {\sum \limits _{r=1}^s{{u}_{r}^{d}\sum \nolimits _{j\ne d}{{\bar{y}}_{rj}}}}+\sum \nolimits _{j\ne d}{\beta _{j}^{y}\left( \Gamma _{j}^{y} \right) }\). Then, the feasible sets of the robust benevolent model and robust aggressive model are the same. \(\square \)

Appenndix B

The random data are presented in Table 6.

Table 6 Dataset of input/output data of 10 DMUs in Section 5.1
Full size table

We present the sensitivity analysis results with different \(\Gamma \) here. Specifically, by fixing \(\Gamma =1, 3\), Tables 7 and 8 list the ratios of fully ranked instances among 500 random instances of the the CCR, RCCR (Sadjadi and Omrani 2008), and proposed RCE-beno model, when varying the number of DMUs.

Table 7 Ratio of fully ranked instances with different number of DMUs (\(\Gamma =1\))
Full size table
Table 8 Ratio of fully ranked instances with different number of DMUs (\(\Gamma =3\))
Full size table

The results of varying the number of input and output, are listed in Tables 9 and 10 (\(\Gamma =1, 3\), respectively). The results show that comparing the the RCCR model (Sadjadi and Omrani 2008), the RCE-beno model has a superior discrimination power.

Table 9 Ratio of fully ranked instances with different input/output combinations (\(\Gamma =1\))
Full size table
Table 10 Ratio of fully ranked instances with different input/output combinations (\(\Gamma =3\))
Full size table

As our second example, we present the efficiencies of 31 Chinese Mainland provincial-level higher education institutes based on the proposed RCE-beno model (Table 11).

Table 11 RCEs of Chinese Mainland 31 provinces’ HEIs
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, J., Shen, L., Zhang, G. et al. Efficiency evaluation with data uncertainty. Ann Oper Res 339, 1379–1403 (2024). https://doi.org/10.1007/s10479-022-04636-0

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-022-04636-0

Keywords

Search

Navigation