A new robust optimization approach to most efficient formulation in DEA

Subject Areas : Data Envelopment Analysis

Reza Akhlaghi ¹ , Mohsen Rostamy-Malkhalifeh ² , Alireza Amirteimoori ³ , Sohrab Kordrostami ⁴

1 - Department of Applied Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran.
2 - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
3 - Department of Applied Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran.
4 - Department of Applied Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.

Received: 2023-12-16 Accepted : 2024-01-09 Published : 2024-01-10

Keywords: Uncertainty, Data Envelopment Analysis (DEA), Robust Optimization, Interval data, Optimistic Counterpart,

Abstract :

In this article, we investigate a new continuous linear model with constraints for the direct selection of the most efficient unit in the analysis of data coverage presented by Akhlaghi et al. (2021) on uncertainty robust optimization. Considering the importance of incorporating uncertainty into performance evaluation models in the real world and its increasing application in various problems, we propose a robust optimization approach. Given the discrete and non-convex nature of the introduced models for selecting the most efficient decision-making unit, examining the dual and finding an optimistic scenario is practically impossible. Therefore, by utilizing the linear model presented by Akhlaghi et al. (2021) with constraints for identifying the most efficient unit, we can investigate the robustness of the desired model using(BS )Bertsimas and Sim's (2004) robust estimation method while also considering uncertainty. We aim to demonstrate that employing a robust formulation leads to reliable performance in uncertain conditions

References:

Akhlaghi, R & Rostamy-Malkhalifeh, M. (2019). A linear programming DEA model for selecting a single efficient unit. International Journal of Industrial Engineering and Operations Research, 1(1), 60-66. http://bgsiran.ir/journal/ojs-3.1.1-4 /index.php/IJIEOR/article/view/12/9.
Akhlaghi, R., Rostamy-Malkhalifeh, M., Amirteimoori, A., & Kordrostami, S. (2021). A Linear Programming Relaxation DEA Model for Selecting a Single Efficient Unit with Variable RTS Technology. Croatian Operational Research Review, 12(2), 131-137.‏
Amin, G. R. (2009). Comments on finding the most efficient DMUs in DEA: An improved integrated model. Computers and Industrial Engineering, 56(4), 1701–1702. https://doi.org/10.1016/j.cie.2008.07.014.
Amin, G. R., & Toloo, M. (2007). Finding the most efficient DMUs in DEA: An improved integrated model. Computers and Industrial Engineering, 52(2), 71–77. https://doi.org/10.1016/j.cie.2006.10.003
Arabmaldar A, Jablonsky J, Hosseinzadeh Saljooghi F. A new robust DEA model and super-efficiency measure. Optimization 2017;66(5):723–36.
Arabmaldar A, Mensah EK, Toloo M. Robust worst-practice interval DEA with non-discretionary factors. Expert Syst Appl 2021:115256.
Baker, R. C., & Talluri, S. (1997). A closer look at the use of data envelopment analysis for technology selection. Computers & Industrial Engineering. Elsevier science ltd. https://doi.org/10.1016/S0360-8352(96)00199-4
Beck, A., & Ben-Tal, A. (2009). Duality in robust optimization: Primal worst equals dual best. Operations Research Letters, 37(1), 1–6. https://doi.org/10.1016/j.orl.2008.09.010
Ben-Tal, A., & Nemirovski, A. (1998). Robust Convex Optimization. Mathematics of Operations Research, 23(4), 769–805. https://doi.org/10.1287/moor.23.4.769
Ben-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)00016-4
Ben-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/PL00011380
Bertsimas D, Sim M. The Price of robustness. Oper Res 2004;52(1):35–53.
Bertsimas, D., Pachamanova, D., & Sim, M. (2004). Robust linear optimization under general norms. Operations Research Letters, 32(6), 510–516. https://doi.org/10.1016/j.orl.2003.12.007
Cazals C, Florens JP, Simar L. Nonparametric frontier estimation: a robust approach. J Econom 2002;106(1):1–25.
Charnes A, Rousseau JJ, Semple JH. Sensitivity and stability of efficiency classifications in data envelopment analysis. J Product Anal 1996;7(1):5–18.
Charnes, 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)90138-8
Cooper WW, Park KS, Yu G. IDEA and AR-IDEA: models for dealing with imprecise data in DEA. Manage Sci 1999;45(4):597–607.
Ehrgott, M., Holder, A., & Nohadani, O. (2018). Uncertain data envelopment analysis. European Journal of Operational Research, 268(1), 231-242.‏
Ertay, T., Ruan, D., & Tuzkaya, U. R. (2006). Integrating data envelopment analysis and analytic hierarchy for the facility layout design in manufacturing systems. Information Sciences 176: 237–262.
Foroughi, A. A. (2011). A new mixed integer linear model for selecting the best decision-making units in data envelopment analysis. Computers and Industrial Engineering, 60(4), 550–554. https://doi.org/10.1016/j.cie.2010.12.012.
Gorissen BL, Yanıko ˘glu ˙I, den Hertog D. A practical guide to robust optimization. Omega (Westport) 2015; 53:124–37.
Hatami-marbini A, Arabmaldar A. Robustness of Farrell cost efficiency measurement under data perturbations: evidence from a US manufacturing application. Eur J Oper Res 2021.
Hibiki N, Sueyoshi T. DEA sensitivity analysis by changing a reference set: regional contribution to Japanese industrial development. Omega (Westport) 1999;27(2):139–53
http://dx.doi.org/10.1016/j.ins.2004.12.001
https://doi.org/10.1016/j.cie.2018.10.006
https://doi.org/10.17535/crorr.2021.0011
Kadziński, M., Labijak, A., & Napieraj, M. (2017). Integrated framework for robustness analysis using ratio-based efficiency model with application to evaluation of Polish airports. Omega, 67, 1-18.‏
Karsak, E. E., & Ahiska, S. S. (2008). Improved common weight MCDM model for technology selection. International Journal of Production Research, 46(24), 6933–6944
Khouja, M. (1995). The use of data envelopment analysis for technology selection. Computers & Industrial Engineering. https://doi.org/10.1016/0360-8352(94)00032-I
Mulvey JM, Vanderbei RJ, Zenios SA. Robust optimization of large-scale systems. Oper Res 1995;43(2):264–81.
Olesen OB, Petersen NC. Chance Constrained efficiency evaluation. Manage Sci 1995;41(3):442–57.
Omrani H, Valipour M, Emrouznejad A. A novel best worst method robust data envelopment analysis: incorporating decision makers’ preferences in an uncertain environment. Oper Res Perspect 2021; 8:100184
Omrani, H., Valipour, M., & Emrouznejad, A. (2021). A novel best worst method robust data envelopment analysis: Incorporating decision makers’ preferences in an uncertain environment. Operations Research Perspectives, 8, 100184.‏
Peykani P, Mohammadi E, Saen RF, Sadjadi SJ, Rostamy-Malkhalifeh M. Data envelopment analysis and robust optimization: a review. Expert Syst 2020(September 2019): e12534.
Sadjadi SJ, Omrani H. A bootstrapped robust data envelopment analysis model for efficiency estimating of telecommunication companies in Iran. Telecomm Policy 2010;34(4):221–32.
Sadjadi SJ, Omrani H. Data envelopment analysis with uncertain data: an application for Iranian electricity distribution companies. Energy Policy 2008;36(11):4247–54
Salahi M, Toloo M, Hesabirad Z. Robust Russell and enhanced Russell measures in DEA. J Oper Res Soc 2019;70(8):1275–83.
Salahi, M., Torabi, N., & Amiri, A. (2016). An optimistic robust optimization approach to common set of weights in DEA. Measurement, 93, 67-73.‏ https://doi.org/10.1080/01605682.2020.1718016
Shang, J., & Sueyoshi, T. (1995). A unified framework for the selection of a Flexible Manufacturing System. European Journal of Operational Research, 85(2), 297–315. https://doi.org/10.1016/0377-2217(94)00041-A
Tavana M, Toloo M, Aghayi N, Arabmaldar A. A robust cross-efficiency data envelopment analysis model with undesirable outputs. Expert Syst Appl 2021; 167:114117.
Toloo, M., & Kresta, A. (2014). Finding the best asset financing alternative: A DEA-WEO approach. Measurement, 55, 288–294.
Toloo, M., & Mensah, E. K. (2018). Robust optimization with nonnegative decision variables: A DEA approach. Computers and Industrial Engineering.
Wang, Y.-M., & Jiang, P. (2012). Alternative mixed integer linear programming models for identifying the most efficient decisionmaking unit in data envelopment analysis. Computers and Industrial Engineering, 62, 546–553.

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A new robust optimization approach to most efficient formulation in DEA

 

R. Akhlaghi 1, M. Rostamy-Malkhalifeh *2, A. Amirteimoori 1, S. Kordrostami 3

1 Department of Applied Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran,

2 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

3 Department of Applied Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran

 

Abstract

In this article, we investigate a new continuous linear model with constraints for the direct selection of the most efficient unit in the analysis of data coverage presented by Akhlaghi et al. (2021) on uncertainty robust optimization. Considering the importance of incorporating uncertainty into performance evaluation models in the real world and its increasing application in various problems, we propose a robust optimization approach. Given the discrete and non-convex nature of the introduced models for selecting the most efficient decision-making unit, examining the dual and finding an optimistic scenario is practically impossible. Therefore, by utilizing the linear model presented by Akhlaghi et al. (2021) with constraints for identifying the most efficient unit, we can investigate the robustness of the desired model using(BS )Bertsimas and Sim's (2004) robust estimation method while also considering uncertainty. We aim to demonstrate that employing a robust formulation leads to reliable performance in uncertain conditions

Keywords: Robust Optimization, Optimistic Counterpart, Uncertainty, Data Envelopment Analysis (DEA), Interval Data.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.       Introduction

Data Envelopment Analysis (DEA) is a popular optimization technique for determining the relative effectiveness of a group of homogeneous Decision-Making Units (DMUs) [3]. The primary objective of certain methods is to select the most efficient DMU rather than to rank DMUs. Therefore, it appears unnecessary to evaluate the performance of each DMU in this condition [4-7]. Consequently, it is essential to directly introduce the most efficient DMU [1,4,8,9]. Small changes in the data have been shown to significantly affect the nominal optimal solution and its feasibility and optimality. This implies that the solution may become meaningless; hence, a practical optimization problem should be modeled and considered an uncertain data problem [10-12]. When there is uncertainty in the data used to determine the solution, there will be some ambiguity regarding the exact values of particular data components. However, in DEA, some data are inherently uncertain. The authors propose several techniques for addressing optimization issues in the presence of uncertainty and perturbed optimization problems, one of which is robust optimization, which has recently gained considerable attention [13,14]. When the underlying data for efficiency analysis is unknown, which is an inevitable feature in real-world scenarios, the classification of DMUs as either efficient or not could be quite misleading. For example, in the financial or aviation industry, accurate performance evaluation is crucial for revenue improvement, as a result, any uncertainty in data could affect management decisions towards the right potential improvement of a DMU or revenue [15,16]. Given that uncertainty a feature of any business environment, and its effect as such is dismissive of performance evaluation, it has become imperative to consider uncertain parameters in the evaluation process and improve the robustness of DEA. Over the years, the issue of robustness, aiming at solutions that are stable to parameter perturbation and preserving the efficiency of DMUs, has been the subject of extensive research by many re-searchers. Research in this direction has been significant because of how robust analysis accounts for uncertainties observed in real-world decision problems [15]. Specifically, three major reasons explain why considering uncertainty and ensuing robust analysis is crucial in DEA. Firstly, the efficiency scores derived for each DMU are obtained by comparing each unit to the other. Therefore, uncertain data could lead to incorrect reference units or the selection of the incorrect best practice unit. In fact, the efficiency or inefficiency of a DMU could be brought to question by a small uncertainty in data. As emphasized in Ehrgott et al. (2018), there is a reasonable argument against the perceived performance of a DMU when the underlying data is imperfect. Secondly, the DEA measures the improvement of the inefficient unit relative to the efficient frontier. This implies that the right amount of potential improvement needed to project an inefficient unit to an efficient one becomes difficult to be measured. Thirdly, the DEA model fails to preserve the efficiency of the DMUs since the model becomes sensitive to a small perturbation in the underlying parameters and data. Several robust techniques have been proposed in the literature to overcome the issue of uncertainty in data. One of the earliest considerations of uncertainty in the mathematical programming community is the use of sensitivity analysis. In DEA, sensitivity analysis dates back to the work of Charnes et al (1978) and focuses on ensuring the stability of the classification of DMUs into efficient and inefficient through preserving the efficiency of DMUs. Algorithmic and distance defining metric techniques are applied to solve this problem, which includes defining a stability region for which data variations will not change a DMU’s classification or omitting an efficient DMU and consequently, changing the reference set for the DMU [18,19]. However, sensitivity analysis is only a post-efficiency analysis because it measures how the efficiency scores with respect to data variation or observations differ from their actual efficiency scores. Therefore, they are not quite an effective measure for robustness [12]. In recent times, stochastic and deterministic approaches that deal with uncertainty in DEA data from the onset and incorporate expert opinions under uncertain environments have been introduced [17,20]. The degree to which the efficiency of DMUs is stable to the underlying uncertainties in the input and output data usually reflects the robustness of the DEA model. The relative efficiency of a DMU is regarded robust when all input and output weights or their representative sample in the DEA models are feasible with respect to its uncertain or imprecise data (2017). Robustness in DEA can be measured with several approaches as mentioned in the previous section. This paper adopts the concept of robustness offered through the lens of robust optimization and applied it to the DEA, known as the RDEA. The robust optimization technique was introduced in Soyster (2021) and developed by Mulvey et al. (1995), Ben-Tal and Nemirovski (1998), and Bertsimas & Sim (2004) among others. The obtained solution of the robust optimization exhibit stability and can withstand changes in the parameters of the model with-out affecting the solution. The reader is referred to Gorissen et al. (2015) for a practical guide to robust optimization. Loosely speaking, a solution under the robust optimization is said to be robust if it is obtained through a robust counterpart an alternative formulation of the nominal optimization problem, that seeks all or most possible realization of the uncertain parameters in an uncertainty set defined by the user. Like robust optimization, the RDEA seeks to similarly immunize uncertain inputs and outputs parameters in a user-defined uncertainty set and provides a probability guarantee for constraint feasibility and reliable performance evaluation, and stable classification of DMUs. Thus, an efficiency of a DMU is said to be ‘robust’ if it’s uncertain inputs and outputs parameters are immunized in an uncertainty set, feasible with respect to the un-certain data and the efficiency is stable to data perturbation. The RDEA method results in an efficiency that is near-optimal efficiency of the nominal DEA and requires no knowledge of the probability distribution of the uncertain input and output data. The RDEA was originally initiated by Sadjadi and Omrani (2008) The authors assumed the existence of uncertainty in output data of DMUs and adapted the robust optimization approach of Ben-Tal and Nemirovski (1999) and Bertsimas & Sim (2004) to correct the efficiency of DMUs. Such robust concept, although similar in objective to the aforementioned approaches (i.e., the statistical-based robust non-parametric estimation methods applied in the work of Cazals et al. (2002) and Daraio and Simar (2020), IDEA of Cooper, Park, & Yu (1999) is quite different due to its immunity against noise, uncertainty, and flexibility in obtaining the robust efficiency. The successful and wide applications of the RDEA method are documented in Peykani et al. (2020). There have been theoretical and practical extensions of the RDEA since the initial model of Sadjadi and Omrani (2008). In Sadjadi and Omrani (2010), the authors combined the RDEA and boot-strapping technique to measure the efficiency of telecommunication companies. Arabmaldar et al. (2017) proposed an RDEA model and robust super-efficiency DEA measures under constant returns to scale (CRS) technology whereas Salahi et al. (2019) proposed a robust Russell measure under interval and ellipsoidal uncertainties in their best and worst-cases. Toloo & Mensah (2018) studied non-negativity conditions in robust optimization and proposed a reduced robust DEA based on variable returns scale (VRS) technology. They applied their model to the efficiency of the largest banks in Europe and showed the computational advantage of the model. Recently, Tavana et al. (2021) developed two DEA adaptations to rank DMUs characterized by interval data and undesirable outputs. They applied their model to assess cross-efficiency and real-life bank data. Considering similar adaptation of interval data and non-discretionary factors, Arabmaldar et al. (2021) proposed a robust worst-practice model to the worst-performing suppliers where some factors in the supply chain decision analysis are not under the discretion of management. Hatami-marbini and Arabmaldar (2021) extended the RDEA to estimate Farrell’s cost efficiency in situations of endogenous and exogenous uncertainties. In the endogenous case, uncertainty in input and output data is assumed whereas exogenous uncertainty is considered for prices of inputs. In the latter case, the robust DEA estimating lower and upper bound of cost efficiencies is given. Although several studies have been done on the RDEA, it still needs further research and development on the methodology.

 

2.       literature review

During the last two decades, data envelopment analysis (DEA) has been widely utilized in many operations research. Karsak and Ahiska (2008) proposed model for finding the most efficient DMUwith a single input and s outputs, Foroughi (2011) proposed two-stage approach to find the most efficient unit and also to fully rank all the DMUs, Toloo& Kresta, (2014) find the most CW-efficient DMU when there are no explicit outputs, Wang and Jiang (2012) proposed MILP model under constant returns to scale (CRS), All the mentioned models are widely used in the deterministic space and some of them are non-linear, but in the real world some phenomena are in the non-deterministic space, for this reason we use the Robust model. Salahi, Torabi, and Amiri (2016) developed a robust counterpart model for the CCR model and established relations between CCR's dual robust counterpart and optimistic robust counterpart. Toloo and Mensah (2018) proposed alternate robust counterparts for nonnegative decision variables. Tavana et al. (2021) developed two DEA adaptations to rank DMUs characterized by interval data and undesir- able outputs. They applied their model to assess cross-efficiency and real-life bank data. Considering similar adaptation of interval data and non-discretionary factors, Arabmaldar et al. (2021) proposed a robust worst practice model to the worst-performing suppliers where some factors in the supply chain decision analysis are not under the discretion of management. Robust DEA (henceforth RDEA) is the application of RO in DEA. The first application of the robust optimization to DEA began in 2008 with Sadjadi & Omrani (2008) when they investigated the performance of utility service providers where the underlying data was uncertain. The authors focused on providing a robust and reliable performance ranking of DMUs for management decision in the utility service. Furthermore, the work of Sadjadi et al (2008), Wang et al (2012) bolstered the need for robust efficiency measure via the RO.

 

3.       Robust DEA

Sadjadi & Omrani [30] were the pioneer researchers that worked on the RDEA model with consideration of uncertainty on output parameters for measuring the performance of Iranian electricity distribution companies. They proposed the robust CCR model based on the robust approaches of Ben-Tal, & Nemirovski. [12,14] Based on Sadjadi, & Omrani [29] study, for considering the uncertainty on outputs, the conventional CCR model that is presented Charnes et all [1] is transformed into (1)

 

Note that each entry  is determined as a symmetric and bounded random variable, which takes the values , where the center of this interval at the point y is a nominal value and  is the perturbation of uncertain parameters . The RDEA model according to robust optimization formulation of Ben-Tal, & Nemirovski. [12] is proposed as (2)

 

As is seen in Model (1,2), to propose RDEA model for dealing with continuous uncertain data, the robust counterpart of all uncertain constraints should be written based on one of the convex uncertainty sets. With respect to this fact that decision maker (DM) can adjust the robustness and linearity of the RDEA model founded upon the Bertsimas & Sim (2004)’s RO method, this model is more effective and applicable in the literature of robust DEA literature.

 

4.       Best efficient unit under uncertainty based on BS approache

This study in deterministic space proposes the continuous linear model proposed by Akhlaghi et al. [2], which is practically superior to other nonlinear models. Applying chaos/disorder to interval data with varying epsilons revealed that optimality remains constant. Initially, we provide a robust counterpart to the LP model shown below to identify the most efficient unit utilizing VRS technology. This is accomplished by replacing binary variables  with a continuous range  for :

Where  denotes the i-th input value for j-th DMU, is the r-th output value for j-th DMU,

represents the weight values of the
r-th output, denotes the weight values of the i-th input,

and  is a non-Archimedean infinitesimal number. by Akhlaghi et al. [2] proposed the most efficient model in deterministic space as follows:

 

The models' optimistic counterparts can also be used to determine the most efficient decision-making unit (DMU) when input and output parameters are uncertain. The efficient decision-making unit is determined optimistically and pessimistically for the best- and worst-case scenarios.

In this section, we investigate a robust and optimistic model under interval uncertainty conditions using the Bertsimas and Sim algorithm to examine the parameters of input and output. As mentioned, uncertainty sets will be considered as intervals. The inputs and outputs of the decision making unit -j denoted as and , respectively belong to the intervals   and . Here,  and represent the nominal or actual values for the input and output, whileand  indicate the maximum deviation. It is clear that the model can take any value within the corresponding interval for inputs and outputs, regardless of the values of other coefficients.

In this approach, the Bertsimas and Sim method is used to express uncertainty, where parameters  and  define the uncertainty budget, representing the maximum non-deterministic parameters for inputs and outputs. belongs to the interval [0, m], and belongs to the interval [0, s]. For  = = 0, the input and output data are the nominal values, and for = s, = m, it means considering the problem with the maximum uncertainty for the problem data.

 

The purpose of introducing the parameters and in the robust formulation is to constrain the problem against uncertainty. Note that here we have:

و 

Therefore, our uncertainty sets are defined as follows:

 

 

 

 

و

 

Now we will use the uncertainty model above to write the  robust form of the problem ( 3)  with Bertsimas and Sim method

Theorem (1): If and belong to the intervals respectively uncertainty  and be, then the  robust  counterpart  (3)  It is equivalent to the following linear problem

(4)

 

 

 

 

Proof: To calculate the  robust  counterpart of (3), we have

 

 

 

(5)

 

now by placing

 

will have

 

 

Therefore, for each we have:

 

So that

 

Since and are positive numbers, so solve the problem

 

It is equivalent to solving the following problem

(6)

 

The double of the problem (6 ) for each is as follows:

 

 

According to the weak duality theorem, for every primal and dual solution we have

 

 

we have for every

(7)

 

Then we always have the following relation

 

 

 

In the same way the problem

 

 

 

Equal to

(8)

 

And also the problem

 

Equal to

(9)

And finally the problem

 

 

Equal to

(10)

 

It is that finally by inserting (7), (8), (9) and (10) in (5) the corresponding  robust  model is obtained .

Theorem (2): If and belonging to uncertainty intervals respectively and be, then the optimistic counterpart (3) It is equivalent to the following linear problem:

 

 

 

 

 

 

 

 

 

Proof: To calculate the optimistic counterpart (3) we have

 

 

 

 

(11)

 

Now, by putting relations (7), (8), (9) and (10) stated in the previous theorem, the optimistic counterpart of the model is obtained.

 

 

 

 

 

 

 

5.       concluding remarks

 

the robust form of the data envelopment analysis models leads to the ranking of the relative efficiency of the decision-making units with a high level of confidence. In addition, it was shown that the robust dual of the BCC form is equivalent to the optimistic counterpart of the BCC form. Another problem of data envelopment analysis is choosing different weights for similar inputs and outputs. Choosing a set of different weights in measuring the efficiency of the units causes the lack of comparability between the efficiency of the units. This study, for reliable performance ranking, using robust pairwise equivalence and optimistic pairwise BCC pairwise equivalence, for uncertain inputs and outputs under interval uncertainty conditions, it was shown that the obtained weights are idea weights. Al are decision makers for each unit. The low volume of calculations compared to similar methods is one of the features of the proposed model. The results show that the use of joint weights obtained from the optimistic peer model provides a more accurate and reliable ranking that is more consistent with reality. This study presented a novel continuous linear model comprising linear bonds. The advantage of the proposed model over previous models is that it is continuous linear; as a result, the model is effectively solved, and its dual problem is calculable and applicable. Furthermore, the dual problem of robust binaries for the linear problem in the presence of span uncertainty was investigated. The study demonstrated that determining the most applicable deciding unit for a given model in the worst-case scenario was equivalent to gauging the most applicable deciding unit for a dual model in the best-case scenario. This paper aimed to discuss the robust counterpart to the new linear programming (LP) model for identifying the most BCC-efficient decision-making unit for interval uncertainty sets. Moreover, it was demonstrated that the robust problem's dual is identical to the dual problem's optimistic counterpart.. While various methods exist to provide solution to inexactness in DEA data (e.g. fuzzy DEA models, Imprecise DEA, Interval DEA, stochastic DEA models), the robust DEA (set-based or scenario-based) set its own unique path in characterizing uncertainty and ensuring probability guarantee for reliable efficiency scores, robust discrimination and ranking of DMUs. At the center of the robust DEA is the robust optimization technique which enables us to model uncertainty in the input and output data of DMUs. For the robust DEA to have impact in theory and application, we feel that methodologies that meet the requirements of computational tractability, guarantee for feasibility of the robust DEA solution in terms of uncertainty in both input and output data and feasibility in probability sense if the uncertainty dynamics obey some natural probability distributions are needed. we focused on the set-based model for uncertainty within the context of robust optimization to advance the modeling of the robust DEA. We propose models which satisfy the robust optimization modeling technique and set the basis for robust DEA modeling and applications.

References

[1]       Charnes, 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)90138-8

 

[2]       Akhlaghi, R., Rostamy-Malkhalifeh, M., Amirteimoori, A., & Kordrostami, S. (2021). A Linear Programming Relaxation DEA Model for Selecting a Single Efficient Unit with Variable RTS Technology. Croatian Operational Research Review, 12(2), 131-137.‏ 

https://doi.org/10.17535/crorr.2021.0011

 

[3]       Bertsimas, D., Pachamanova, D., & Sim, M. (2004). Robust linear optimization under general norms. Operations Research Letters, 32(6), 510–516. https://doi.org/10.1016/j.orl.2003.12.007

 

[4]       Amin, G. R. (2009). Comments on finding the most efficient DMUs in DEA: An improved integrated model. Computers and Industrial Engineering, 56(4), 1701–1702. https://doi.org/10.1016/j.cie.2008.07.014.

 

[5]       Foroughi, A. A. (2011). A new mixed integer linear model for selecting the best decision-making units in data envelopment analysis. Computers and Industrial Engineering, 60(4), 550–554. https://doi.org/10.1016/j.cie.2010.12.012.

 

[6]       Khouja, M. (1995). The use of data envelopment analysis for technology selection. Computers & Industrial Engineering. https://doi.org/10.1016/0360-8352(94)00032-I

 

[7]       Salahi, M., Torabi, N., & Amiri, A. (2016). An optimistic robust optimization approach to common set of weights in DEA. Measurement, 93, 67-73.‏ https://doi.org/10.1080/01605682.2020.1718016

 

[8]       Akhlaghi, R & Rostamy-Malkhalifeh, M. (2019). A linear programming DEA model for selecting a single efficient unit. International Journal of Industrial Engineering and Operations Research, 1(1), 60-66. http://bgsiran.ir/journal/ojs-3.1.1-4 /index.php/IJIEOR/article/view/12/9.

 

[9]       Shang, J., & Sueyoshi, T. (1995). A unified framework for the selection of a Flexible Manufacturing System. European Journal of Operational Research, 85(2), 297–315. https://doi.org/10.1016/0377-2217(94)00041-A

 

[10]       Beck, A., & Ben-Tal, A. (2009). Duality in robust optimization: Primal worst equals dual best. Operations Research Letters, 37(1), 1–6. https://doi.org/10.1016/j.orl.2008.09.010

[11]       Ben-Tal, A., & Nemirovski, A. (1998). Robust Convex Optimization. Mathematics of Operations Research, 23(4), 769–805. https://doi.org/10.1287/moor.23.4.769

 

[12]       Ben-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)00016-4

 

[13]       Baker, R. C., & Talluri, S. (1997). A closer look at the use of data envelopment analysis for technology selection. Computers & Industrial Engineering. Elsevier science ltd. https://doi.org/10.1016/S0360-8352(96)00199-4 

 

[14]       Ben-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/PL00011380

 

[15]       Kadziński, M., Labijak, A., & Napieraj, M. (2017). Integrated framework for robustness analysis using ratio-based efficiency model with application to evaluation of Polish airports. Omega, 67, 1-18.‏

 

[16]       Omrani, H., Valipour, M., & Emrouznejad, A. (2021). A novel best worst method robust data envelopment analysis: Incorporating decision makers’ preferences in an uncertain environment. Operations Research Perspectives, 8, 100184.‏

 

[17]       Ehrgott, M., Holder, A., & Nohadani, O. (2018). Uncertain data envelopment analysis. European Journal of Operational Research, 268(1), 231-242.‏

 

[18]       Charnes A, Rousseau JJ, Semple JH. Sensitivity and stability of efficiency classifications in data envelopment analysis. J Product Anal 1996;7(1):5–18.

 

[19]       Hibiki N, Sueyoshi T. DEA sensitivity analysis by changing a reference set: regional contribution to Japanese industrial development. Omega (Westport) 1999;27(2):139–53

 

[20]       Omrani H, Valipour M, Emrouznejad A. A novel best worst method robust data envelopment analysis: incorporating decision makers’ preferences in an uncertain environment. Oper Res Perspect 2021; 8:100184

 

[21]       Mulvey JM, Vanderbei RJ, Zenios SA. Robust optimization of large-scale systems. Oper Res 1995;43(2):264–81.

 

[22]       Bertsimas D, Sim M. The Price of robustness. Oper Res 2004;52(1):35–53.

 

[23]       Gorissen BL, Yanıko ˘glu ˙I, den Hertog D. A practical guide to robust optimization. Omega (Westport) 2015; 53:124–37.

 

[24]       Sadjadi SJ, Omrani H. Data envelopment analysis with uncertain data: an application for Iranian electricity distribution companies. Energy Policy 2008;36(11):4247–54

 

[25]       Cazals C, Florens JP, Simar L. Nonparametric frontier estimation: a robust approach. J Econom 2002;106(1):1–25.

 

[26]       Olesen OB, Petersen NC. Chance Constrained efficiency evaluation. Manage Sci 1995;41(3):442–57.

 

[27]       Cooper WW, Park KS, Yu G. IDEA and AR-IDEA: models for dealing with imprecise data in DEA. Manage Sci 1999;45(4):597–607.

 

[28]       Peykani P, Mohammadi E, Saen RF, Sadjadi SJ, Rostamy-Malkhalifeh M. Data envelopment analysis and robust optimization: a review. Expert Syst 2020(September 2019): e12534.

 

[29]       Sadjadi SJ, Omrani H. A bootstrapped robust data envelopment analysis model for efficiency estimating of telecommunication companies in Iran. Telecomm Policy 2010;34(4):221–32.

 

[30]       Arabmaldar A, Jablonsky J, Hosseinzadeh Saljooghi F. A new robust DEA model and super-efficiency measure. Optimization 2017;66(5):723–36.

 

[31]       Salahi M, Toloo M, Hesabirad Z. Robust Russell and enhanced Russell measures in DEA. J Oper Res Soc 2019;70(8):1275–83.

 

[32]       Toloo, M., & Mensah, E. K. (2018). Robust optimization with nonnegative decision variables: A DEA approach. Computers and Industrial Engineering.

https://doi.org/10.1016/j.cie.2018.10.006

 

[33]       Tavana M, Toloo M, Aghayi N, Arabmaldar A. A robust cross-efficiency data envelopment analysis model with undesirable outputs. Expert Syst Appl 2021; 167:114117.

 

[34]       Arabmaldar A, Mensah EK, Toloo M. Robust worst-practice interval DEA with non-discretionary factors. Expert Syst Appl 2021:115256.

 

[35]       Hatami-marbini A, Arabmaldar A. Robustness of Farrell cost efficiency measurement under data perturbations: evidence from a US manufacturing application. Eur J Oper Res 2021.

 

[36]       Amin, G. R., & Toloo, M. (2007). Finding the most efficient DMUs in DEA: An improved integrated model. Computers and Industrial Engineering, 52(2), 71–77. https://doi.org/10.1016/j.cie.2006.10.003

 

[37]       Ertay, T., Ruan, D., & Tuzkaya, U. R. (2006). Integrating data envelopment analysis and analytic hierarchy for the facility layout design in manufacturing systems. Information Sciences 176: 237–262.

http://dx.doi.org/10.1016/j.ins.2004.12.001

 

[38]       Karsak, E. E., & Ahiska, S. S. (2008). Improved common weight MCDM model for technology selection. International Journal of Production Research, 46(24), 6933–6944

 

[39]       Toloo, M., & Kresta, A. (2014). Finding the best asset financing alternative: A DEA-WEO approach. Measurement, 55, 288–294.

 

[40]       Wang, Y.-M., & Jiang, P. (2012). Alternative mixed integer linear programming models for identifying the most efficient decisionmaking unit in data envelopment analysis. Computers and Industrial Engineering, 62, 546–553.

 

 

 

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