Effect of Returns to Scale in Different DEA Models on Evaluating Efficiency by Considering Uncertainty in Data: Application from HospitalsShayanArefNajafiSeyyed EsmaeelAhadzadeh NaminMahnazDepartment of Industrial Engineering, Islamic Azad University, Saveh Science and Research Branch, Iran.Department of Industrial Engineering, Islamic Azad University, Science and Research Branch,Tehran, Iran.Department of Mathematics, Shahr-e-Qods Branch, Islamic Azad University, Tehran, Iran.Problem Statement
DEA is a nonparametric linear programming technique where a set of units is evaluated according to their input consumption and output production [9] since the seminal paper by Charnes et al. [9], a variety of DEA models have appeared in the literature. Two of the DEA models that are most often associated with the DEA methodology are the CCR and BCC models.
Let xij be the inputs for a decision unit with i=1,…,m and yrj be the outputs with r=1,…,s and j=1,…,n. Let ui and vj be the dual variables associated with xi and yj, respectively. The linear programming statement for the (Input-oriented) CCR model is:
max z = ∑ ur yro ,
s m (1) ∑ ur yrj − ∑ vi xij ≤ 0, j = 1, … , n,
r=1 i=1
ur, vi ≥ 0.
The dual (Multiplier) form of the BCC model is formulated as follows [10]:
max z = ∑ ur yro + d0,
s m (2) ∑ ur yrj − ∑ vi xij + d0 ≤ 0,
r=1 i=1
ur, vi ≥ ε, d0 is free.
When there is uncertainty with the inputs and the outputs, we must use different techniques to make sure that a small change in input/output data does not change the output rankings. Thus, we use the RDEA to handle the uncertainty of data.
Robust Optimization
In classical optimization modeling, input parameters are considered as certain values. However, in real cases, we are not certain about all parameter values. Robust optimization is a new approach to incorporate uncertainty within mathematical models. The approach based on Robust optimization is the most preferred method among practitioners due to its applicability. Recently, Robust optimization is very popular among practitioners and is applied in different contexts [7].
In classical modeling, a full probabilistic characterization is assumed under uncertainty. However, a representative nominal value is used instead of uncertainty, which is ignored in many models. Stochastic Programming (SP) is the classical approach to handling uncertainty. Recently, Robust optimization has been developed, which is considered an alternative to sensitivity analysis and SP [8].
Ben-Tal and Nemirovski [2] and Bertsimas and Sim [11] to handle uncertainty in data created new Robust optimization approaches. To present the Robust structure proposed by Ben-Tal and Nemirovski [2] and Bertsimas and Sim [11], assume the following LP problem:
min c′x,
Ax ≥ b,
xϵx.
By considering that the uncertainty influences the constraints A, in a Robust optimization approach, for expressing uncertainty in data, we consider a particular row i of the matrix A and assume the Ji is the set of coefficients in row i that lead to uncertainty in the data. Each input data by an uncertainty value, (a~ ij) is expressed as follows:
(a~ ij) take values in [aij -a^ ij , aij +a^ ij], aij is the nominal value and a^ ij is called the precision of the estimate. In the next section, express the RDEA based on Bertsimas and Sim's [5, 11, 12] and Bertsimas et al. approach.
2.2|Robust Data Envelopment Analysis Based on Bertsimas and Sim [5, 11, 12] and Bertsimas et al. approach
By considering Ji: Set of coefficients of uncertain data, ith row constraint, Model (2) reformulated as follows: min c′x,
a^ ijx≥b,
x ϵ x.
Bertsimas and Thiele [13], [14] measures the deviation of parameter a~ ij from aij as follows:
a~ ij−aij
ηij = a^ , (5)
ij
a~ ij and aij are uncertain data and a nominal value, and a^ ij measures the precision of the estimate. ηij has an unknown but symmetric distribution that takes values in [-1,1]. Thus,
∑n j=1 ηij is in [−n, n], ∑n j=1 ηij ≤ Γi, Γi is called the budget of uncertainty of constraint i and takes a value in [0, n].
Γi adjust the Robustness of the proposed method against the level of conservatism of the solution.
There are three cases for Γi as follows:
I. If Γi=0, there is no protection against uncertainty.
II. If Γi=0, the ith constraint of the problem is completely protected against uncertainty.
III. If Γiϵ (0,n), the decision-maker makes a trade-off between the protection level of the constraint and the degree of conservatism of the solution [4].
It is sufficient to choose Γi at least equal to Γi=1+Φ−1 (1-ei )√n .
n is the number of uncertain parameters, and Φ is the CDF of a Gaussian distribution. However, for the case of our proposed method, since there are only four uncertain parameters, we choose Γ = 4 as recommended by Sadjadi and Omrani [4].
The Robust optimization based on Bertsimas and Sim [5], [11], [12], and Bertsimas and Sim [11] is as follows:
min c′x,
a′ ix − Γi pi − ∑jϵJi qij ≥ 0,
pi + qij ≥ eaij yj,
−yj ≤ xj ≤ yj, (6)
pi, qij ≥ 0,
xϵx.
By considering Model (1) as linear programming and for uncertainty in outputs, use Model (6) and write the RDEA model based on Bertsimas et al.'s approach as follows [4]:
max z, ∑ ur yro − z0 − Γ0 p0 − ∑ qro ≥ 0,
r=1 jϵJi
∑ vi xij − ∑ ur yrj − Γj pj − ∑ qrj ≥ 0, j = 1, … . . , n, i=1 r=1 jϵJ (7)
pj + qrj ≥ eyrj zr , for all r, j,
−zr ≤ ur ≤ zr, for all r,
pj, qrj ≥ 0,
vi, ur ≥ 0.
By considering Model (2) as linear programming and for uncertainty in outputs, we used Model (6) to write the RDEA model based on Bertsimas et al.'s approach as follows:
max z,
∑ ur yro − z0 − Γ0 p0 − ∑ qro + d0 ≥ 0,
r=1 jϵJi
∑ vi xij − ∑ ur yrj − Γj pj − ∑ qrj − d0 ≥ 0, j = 1, … . . , n, i=1 r=1 jϵJi (8)
pj + qrj ≥ eyrj zr , for all r, j,
−zr ≤ ur ≤ zr , for all r,
pj, qrj ≥,
vi, ur ≥ 0.
In Models (7) and (8), xio and yro are the ith input and rth output for the DMU under consideration. xij and yrj are the ith input and rth output for the company j, also z0 is the efficiency for the DMU under consideration. In the RDEA, ei is the most probable for violating the constraint i that takes value in [ 0.001,0.1]. pj, qrj are the dual variables. We used the Software GAMS for solving these models.
Case Study
In order to present a more detailed explanation of the above models, we used the actual data and compared the obtained results. In this paper, we considered 39 health centers and hospitals for evaluating and ranking them based on the CCR, BCC, and RDEA models. In Table 1, we show the inputs and outputs of 39 health centers and hospitals that are certain, and by considering uncertainty in data, we evaluate and rank them. Our aim in this paper is to check the effect of return to scale on evaluating efficiency and ranking them on the application of CCR and BCC models, and RDEA based on the CCR and BCC models of DEA. So in Table 3, we show the obtained results of evaluating efficiency and ranking the health centers and hospitals by considering certain and uncertain data based on the CCR model, and in Table 4, we show the obtained results of evaluating efficiency and ranking the health centers and hospitals by considering certain and uncertain data based on the BCC model.
Input and output of the health centers.
This table is shown in the PDF version of the article.
The inputs and the outputs of the Robust data envelopment analysis model.
This table is shown in the PDF version of the article.
ranked the health centers and hospitals.
Table 2 shows the inputs and the outputs of the RDEA model that we used for considering uncertainty in data. In this paper, we considered the uncertainty in output data and, by using Models (1)-(8), evaluated and In the Column 2 of Table 3 by using the CCR model of DEA we evaluated and ranked the health centers and hospitals that shows centers 19, 23, 13,13, 20, 21, 35, 11, 26, 14, 35 are efficient DMUs and their degree of efficiency are one and then by using the A and P technique [15] respectively have the highest degree of efficiency. The obtained results by using the RDEA for controlling the uncertainty in output data show that the ranking of the efficient DMUs changed and led to a reduction in the degree of efficiency of DMUs. In general, using the RDEA reduces the degree of efficiency of DMUs, and by increasing the perturbation level (e) from 0.001 to 0.1, the degree of efficiency of DMUs is reduced. For example, Imam Khomeini Hospital Dargaz's degree of efficiency is 1 in the crisp model of CCR, but in our Robust model, the degree of efficiency is reduced to about 0.793699 when the level of uncertainty for all output parameters is 0.1 .
Evaluating and Ranking based on the CCR model.
This table is shown in the PDF version of the article.
Evaluating and Ranking based on the BCC model.
This table is shown in the PDF version of the article.
In the Column 2 of Table 4 by using the BCC model of DEA we evaluated and ranked the health centers and hospitals that shows centers 19, 23, 13, 20, 21, 35, 22, 11, 26, 14, 3, 36, 18 are efficient DMUs and their degree of efficiency are one and then by using the A and P technique [15] respectively have the highest degree of efficiency. The obtained results by using the RDEA for controlling the uncertainty in output data show that the rank of the efficient DMUs changed and led to a reduction in the degree of efficiency of DMUs. In general, by using the RDEA, the degree of efficiency of DMUs decreased, and by increasing the perturbation level (e) from 0.001 to 0.1, the degree of efficiency of DMUs decreased. For example, Imam Khomeini Hospital Dargaz's degree of efficiency is 1 in the crisp model of BCC, but in our Robust model, the degree of efficiency is reduced to about 0.929111 when the level of uncertainty for all output parameters is 0.1. This fact shows that the precision of estimated data has a significant influence on the results of efficiency measurement. By using statistical tests based on Spearman Pearson [16] for comparing the obtained results from the DEA and RDEA ranking, as follows:
Where n is the number of units and di is calculated as the difference between the rankings of DEA and Robust DEA [6].
6 ∑ d2 i
dspearman=1- 2=1 .
n(n −1)
By using the results of Table 3, dspearman= 0.976518, which means, statistically, there is no significant difference between the DEA and RDEA ranking in the CCR model of DEA.
By using the results of Table 4, dspearman=0.969636, which means, statistically, there is no significant difference between the DEA and RDEA ranking in the BCC model of DEA.
By using the results of Table 3 and Table 4, dspearman=0.924899, which means, statistically, there is no significant difference between the ranking of CCR and the BCC model of DEA.
By using the results of Table 3 and Table 4, dspearman=0.8848817, which means, statistically, there is no significant difference between the ranking of RDEA based on the CCR and BCC model.
Conclusion
Considering the results obtained with this setup, which reached, though there is a difference in ranking DMUs between CRS and the VRS approaches, there is no significant difference between the ranking of RDEA based on the CCR and BCC models, and there is no significant difference between the ranking of the BCC and CCR models of DEA. Results show that considering the uncertainty in the data can lead to a loss of efficiency scores. And by increasing the perturbation level (e) from 0.001 to 0.1, the efficiency of DMUs was reduced.
ReferencesBen-Tal, A., & Nemirovski, A. (1999). Robust solutions of uncertain linear programs. Operations research letters, 25(1), 1–13. https://doi.org/10.1016/S0167-6377(99)00016-4Ben-Tal, A., & Nemirovski, A. (2000). Robust solutions of linear programming problems contaminated with uncertain data. Mathematical programming, 88(3), 411–424. https://doi.org/10.1007/PL00011380Sadjadi, S. J., & Omrani, H. (2010). A bootstrapped robust data envelopment analysis model for efficiency estimating of telecommunication companies in Iran. Telecommunications policy, 34(4), 221–232. https://doi.org/10.1016/j.telpol.2009.09.003Sadjadi, S. J., & Omrani, H. (2008). Data envelopment analysis with uncertain data: An application for Iranian electricity distribution companies. Energy policy, 36(11), 4247–4254. https://doi.org/10.1016/j.enpol.2008.08.004Bertsimas, D., & Sim, M. (2003). Robust discrete optimization and network flows. Mathematical programming, 98(1), 49–71. https://doi.org/10.1007/s10107-003-0396-4Roghanian, E., & Foroughi, A. (2010). An empirical study of Iranian regional airports using robust data envelopment analysis. International journal of industrial engineering computations, 1(1), 65–72. https://doi.org/10.5267/j.ijiec.2010.01.006Khaki, A. R., Sadjadi, S. J., Gharakhani, M., & Rashidi, S. (2012). Data envelopment analysis under uncertainty: A case study from public healthcare. African journal of business management, 6(24), 7096. https://doi.org/10.5897/AJBM11.591Gharakhani, M., Kazemi, I., & Alizadeh Haji, H. (2011). A robust DEA model for measuring the relative efficiency of Iranian high schools. Management science letters, 1(3), 389–404. https://doi.org/10.5267/j.msl.2011.01.002Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European journal of operational research, 2(6), 429–444. https://doi.org/10.1016/0377-2217(78)90138-8Banker, R. D., Charnes, A., & Cooper, W. W. (1984). Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management science, 30(9), 1078–1092. https://doi.org/10.1287/mnsc.30.9.1078Bertsimas, D., & Sim, M. (2004). The price of robustness. Operations research, 52(1), 35–53. https://doi.org/10.1287/opre.1030.0065Bertsimas, D., & Sim, M. (2006). Tractable approximations to Robust conic optimization problems. Mathematical programming, 107(1), 5–36. https://doi.org/10.1007/s10107-005-0677-1Bertsimas, D., & Thiele, A. (2006). A robust optimization approach to inventory theory. Operations research, 54(1), 150–168. https://doi.org/10.1287/opre.1050.0238Bertsimas, D., & Thiele, A. (2006). Robust and data-driven optimization: Modern decision making under uncertainty. In Models, methods, and applications for innovative decision making (pp. 95–122). INFORMS. https://doi.org/10.1287/educ.1063.0022Andersen, P., & Petersen, N. C. (1993). A procedure for ranking efficient units in data envelopment analysis. Management science, 39(10), 1261–1264. https://doi.org/10.1287/mnsc.39.10.1261Maritz, J. S. (1995). Distribution-free statistical methods. Taylor & Francis. https://doi.org/10.1201/9781003059905