A Combined OCBA–AIC Method for Stochastic Variable Selection in Data Envelopment Analysis

Abstract

This study introduces a novel approach to enhance variable selection in Data Envelopment Analysis (DEA), especially in stochastic environments where efficiency estimation is inherently complex. To address these challenges, we propose a game cross-DEA model to refine efficiency estimation. Additionally, we integrate the Akaike Information Criterion (AIC) with the Optimal Computing Budget Allocation (OCBA) technique, creating a hybrid method named OCBA–AIC. This innovative method efficiently allocates computational resources for stochastic variable selection. Our numerical analysis indicates that OCBA–AIC surpasses existing methods, achieving a lower AIC value. We also present two real-world case studies that demonstrate the effectiveness of our approach in ranking suppliers and tourism companies under uncertainty by selecting the most suitable partners. This research enriches the understanding of efficiency measurement in DEA and makes a substantial contribution to the field of performance management and decision-making in stochastic contexts.

1. Introduction

Data Envelopment Analysis (DEA) is a prominent method for assessing the efficiency of decision-making units (DMUs) [1]. Critical to its success is the selection of appropriate inputs and outputs, a task that poses significant challenges for decision-makers, particularly in stochastic environments where data uncertainties can significantly affect efficiency estimates [2,3].

Despite DEA’s widespread adoption, variable-selection methods often fall short. The lack of standardized procedures has led analysts to rely heavily on subjective experiences [4]. Existing techniques, while helpful, may lack the robustness and precision required for stochastic data, frequently underestimating the impact of randomness on efficiency assessments [5]. Recent studies have highlighted the pivotal role of variable selection in enhancing DEA’s accuracy. Innovations such as machine learning for variable selection [6], a three-stage DEA for electric utility modeling [7], and mathematical frameworks for feature selection [8] have contributed to the field. Additionally, the role of variable selection in port efficiency evaluation has been insightfully discussed [9].

In response to these challenges, we introduce a novel game cross-DEA model to refine efficiency estimation [10]. We also present a pioneering method that integrates Optimal Computing Budget Allocation (OCBA) with the Akaike Information Criterion (AIC) for stochastic variable selection [11]. This integration is designed to navigate the complexities of variable selection amid randomness, offering innovative solutions to the field.

By incorporating advanced analytics and statistical techniques like AIC, this study bolsters the robustness of DEA models, leading to more precise and reliable efficiency measurements [12]. AIC’s role in model selection ensures the choice of the most suitable model based on its goodness of fit and simplicity, therefore enhancing the decision-making process [13].

The implications of this research are profound for DMUs operating in dynamic and uncertain environments. The capacity to make informed decisions based on a comprehensive evaluation of variables and their interactions is essential for strategic planning and resource allocation [14]. The strategic role of decision-making, particularly in aligning organizational goals with resource management, is well-established [15]. This study, therefore, not only deepens the theoretical understanding of DEA in stochastic settings but also offers tangible benefits to decision-makers, equipping them with tools to navigate complexity and uncertainty in their decision-making processes.

This paper makes three key contributions to the literature. First, it enhances DEA by integrating self and peer evaluations for a more nuanced assessment of DMUs. Second, it advances stochastic DEA with a novel model that combines game cross-efficiency, entropy, and AIC, improving efficiency measurement accuracy amid data uncertainties [16]. Third, it refines the stochastic DEA model by [11] through OCBA, optimizing computational resources and enhancing model efficacy in stochastic contexts [13]. Collectively, these contributions provide a robust variable-selection approach for DEA, offering both theoretical enrichment and practical tools for decision-making [17,18]. Our methodology supports strategic and operational planning and aligns organizational objectives with resource management, addressing the complexities of the modern business environment [19].

The paper is structured as follows: Section 2 offers an extensive literature review, critiquing current methodologies and their shortcomings. Section 3 elaborates on our proposed methodology. Section 4 provides a numerical example to demonstrate our method’s variable-selection prowess. Section 5 features two real-world case studies applying our approach, and Section 6 concludes the paper, encapsulating our contributions and proposing avenues for future inquiry.

2. Literature Review

Data Envelopment Analysis (DEA), introduced by [20], is a non-parametric method for assessing the relative efficiency of decision-making units (DMUs). Since then, DEA has evolved substantially, with key models such as the CCR model and the BCC model being widely implemented across various sectors, including healthcare [19], banking [1], and manufacturing [21]. These models operate under the assumption of constant and variable returns to scale, respectively, providing a robust framework for efficiency measurement [22].

The selection of appropriate input and output variables in DEA is crucial since it directly affects the efficiency scores and interpretations of the analysis. Currently, there are no standardized guidelines for selecting optimal inputs and outputs. Friedman and Sinuany-Stern [23] proposed a general rule of thumb suggesting that the total number of variables should not exceed one third of the total DMUs. Norman and Stoker [24] developed a model to assess the correlation of efficiencies when adding variables to the DEA model, finding that efficiency can be significantly affected by a variable if a high statistical correlation is present. Jenkins and Anderson [17] utilized regression and partial correlation methods to determine which variables should be omitted; however, these approaches can yield unstable results even under similar conditions, necessitating caution in selecting suitable methods for identifying model variables [4].

A comprehensive summary of the literature on variable selection in DEA is provided in Table 1. The table highlights various models, key findings, and the types of methodologies employed across different years. As shown, each study contributes uniquely to the discourse, emphasizing either specific techniques for variable selection or broader guidelines, therefore illuminating the evolving nature of DEA methodology.

Wagner and Shimshak [2] employed a stepwise selection approach to analyze changes in efficiencies, though this method does not account for inter-variable relationships and risks losing important information. Each of these techniques has strengths; for example, stepwise methods can simplify model complexity, but they may overlook critical interactions among variables [25].

Table 1. Summary of Literature on Variable Selection in DEA.

Reference	Model	Key Finding	Type	Year
[23]	Ranking scales	Variable selection in DEA context	Static	1998
[17]	Statistical approach	Reduces variables in DEA	Static	2003
[2]	Stepwise selection	Procedures for variable selection	Static	2007
[4]	Variable-selection techniques	Guidelines for selection methods	Static	2011
[26]	Multicriteria selection	Classifies production batches	Static	2012
[27]	cross-efficiency	Interval DEA	Stochastic	2012
[5]	Alternative approaches	Alternative performance measures	Static	2015
[28]	cross-efficiency	supplier selection	Static	2015
[11]	Akaike’s criteria	Variable selection via AIC	stochastic	2017
[9]	Stepwise selection	Port efficiency assessment	Static	2020
[29]	Variable selection	Overview of variable selection	Static	2020
[30]	Relevance measures	Critical values methodology	Static	2020
[7]	Virtual frontier	Efficiency modeling	Static	2021
[31]	Stepwise method	New method for variable selection	Static	2021
[32]	Entropy measures	Novel selection method	Static	2021
[6]	Machine learning	Variable selection in DEA	Static	2023

Table 1. Summary of Literature on Variable Selection in DEA.

Reference	Model	Key Finding	Type	Year
[23]	Ranking scales	Variable selection in DEA context	Static	1998
[17]	Statistical approach	Reduces variables in DEA	Static	2003
[2]	Stepwise selection	Procedures for variable selection	Static	2007
[4]	Variable-selection techniques	Guidelines for selection methods	Static	2011
[26]	Multicriteria selection	Classifies production batches	Static	2012
[27]	cross-efficiency	Interval DEA	Stochastic	2012
[5]	Alternative approaches	Alternative performance measures	Static	2015
[28]	cross-efficiency	supplier selection	Static	2015
[11]	Akaike’s criteria	Variable selection via AIC	stochastic	2017
[9]	Stepwise selection	Port efficiency assessment	Static	2020
[29]	Variable selection	Overview of variable selection	Static	2020
[30]	Relevance measures	Critical values methodology	Static	2020
[7]	Virtual frontier	Efficiency modeling	Static	2021
[31]	Stepwise method	New method for variable selection	Static	2021
[32]	Entropy measures	Novel selection method	Static	2021
[6]	Machine learning	Variable selection in DEA	Static	2023

To refine variable selection, Toloo and Tichy [5] introduced a model that determines the number of variables by solving a series of mixed-integer linear programming (MILP) problems. However, it remains unable to specifically identify which variables should be selected, as it primarily considers the frequency of variable occurrence.

Furthermore, the issue of stochastic data significantly impacts the reliability of efficiency results. In real-world scenarios, various factors contribute to uncertainty, such as behavioral biases, technical malfunctions, and human errors, often resulting in data with stochastic characteristics [3]. This longstanding issue continues to obstruct efficiency measurement and performance evaluation [33].

Several methodologies have been developed to address these challenges. Despotis and Smirlis [33] treated efficiency as an interval and introduced the concept of interval efficiency for measurement, which is effective only when extreme values are out of range, as differing degrees of intervals can lead to varying levels of uncertainty [34]. Kao and Liu [35] investigated interval efficiency and its impact on efficiency scores, measuring data uncertainty through simulations with specific distributions. However, results may lack robustness when estimating efficiency over multiple periods, as technological changes can significantly influence performance, particularly regarding the identification of the best-performing DMU.

Wong et al. [18] combined Monte Carlo simulations with the Optimal Computing Budget Allocation (OCBA) model [16] for efficiency measurement in stochastic environments. The OCBA technique offers a novel approach to efficiently allocate computational resources in the context of simulation-based optimization. Its innovative features include the ability to balance the trade-off between exploration and exploitation, thus maximizing the accuracy of efficiency estimates while minimizing computational costs [16]. By intelligently selecting the most promising simulations based on early performance indicators, OCBA enhances the robustness of efficiency measurement, especially in complex and stochastic environments. Utilizing OCBA not only streamlines the simulation process but also significantly reduces the overall data collection burden [18]. This is particularly advantageous in efficiency studies where time and resource constraints are common. The innovative aspect of OCBA lies in its adaptive nature, allowing for real-time adjustments based on the performance of the evaluated DMUs.

In conclusion, the literature on DEA is replete with innovative methodologies that have evolved to address the complexities of efficiency measurement. The integration of OCBA with DEA, as well as the application of statistical techniques for variable selection, signifies a significant leap forward in the field. As we continue to navigate the challenges posed by stochastic data and the need for precise variable selection, the contributions of [5,11,18,20,22,33,35] among others, provide a solid foundation for future research and application in DEA.

This study aims to contribute to the literature by leveraging the OCBA technique to determine the optimal number of simulation replications and select the best combination of input and output variables. By doing so, we seek to provide a robust methodology for variable selection in DEA applications, therefore advancing both theoretical understanding and practical implications for decision-makers [16].

3. Our Proposed Method

3.1. Stochastic DEA

In a system with multiple inputs and outputs, consider n independent decision-making units (DMUs), each equipped with m inputs and s outputs. For DMU j, let $x_{ij}$ denote its i-th input and $y_{rj}$ its r-th output, where $i\in \{1,\dots ,m\}$ and $r\in \{1,\dots ,s\}$. The efficiency score for any DMU d can be calculated by solving the following CCR model:

$$\begin{array}{c}{E}_{dd}=\max {\displaystyle \sum _{r=1}^s}{u}_{rd}{y}_{rd}\hfill \\ \mathrm{s}.\mathrm{t}.\hfill \\ {\displaystyle \sum _{r=1}^s}{u}_{rd}{y}_{rj}-{\displaystyle \sum _{i=1}^m}{v}_{id}{x}_{ij}\le 0,j=1,\dots ,n\hfill \\ {\displaystyle \sum _{i=1}^m}{v}_{id}{x}_{id}=1\hfill \\ u_{rd}\ge 0,r=1,\dots ,s,\hfill \\ v_{id}\ge 0,i=1,\dots ,m.\hfill \end{array}$$

(1)

For each DMU d within the evaluation set $\{1,2,\dots ,n\}$, the optimal weights $u_{rd}^{*}$ and $v_{id}^{*}$ are derived by solving the aforementioned model. This model, labeled as Equation (1), is a static frontier model that assumes all inputs and outputs are deterministic. When accounting for the stochastic characteristics of the variables, as discussed by [36], the input and output variables are denoted by ${\tilde{x}}_{ij}\sim N({\overline{x}}_{ij},{\delta }_{x_{ij}}^2)$ and ${\tilde{y}}_{rj}\sim N({\overline{y}}_{rj},{\delta }_{y_{rj}}^2)$, respectively. Consequently, the DEA model transforms into the following stochastic DEA programming model:

$$\begin{array}{c}{\tilde{E}}_{dd}=\max {\displaystyle \sum _{r=1}^s}{u}_{rd}{\tilde{y}}_{rd}\hfill \\ \mathrm{s}.\mathrm{t}.\hfill \\ {\displaystyle \sum _{r=1}^s}{u}_{rd}{\tilde{y}}_{rj}-{\displaystyle \sum _{i=1}^m}{v}_{id}{\tilde{x}}_{ij}\le 0,j=1,\dots ,n\hfill \\ {\displaystyle \sum _{i=1}^m}{v}_{id}{\tilde{x}}_{id}=1\hfill \\ u_{rd}\ge 0,r=1,\dots ,s,\hfill \\ v_{id}\ge 0,i=1,\dots ,m.\hfill \end{array}$$

(2)

Based on the game model proposed by [21], the aggressive and benevolent game cross-efficiencies are defined as follows:

$$\begin{array}{c}{E}_{dj}^L=\min {\displaystyle \sum _{r=1}^s}{u}_{rd}{\tilde{y}}_{rj}\hfill \\ \mathrm{s}.\mathrm{t}.\hfill \\ {\displaystyle \sum _{r=1}^s}{u}_{rd}{\tilde{y}}_{rd}-E_{dd}{\displaystyle \sum _{i=1}^m}{v}_{id}{\tilde{x}}_{id}=0\hfill \\ {\displaystyle \sum _{r=1}^s}{u}_{rd}{\tilde{y}}_{rj}-{\displaystyle \sum _{i=1}^m}{v}_{id}{\tilde{x}}_{ij}\le 0,j=1,\dots ,n,j\ne d\hfill \\ {\displaystyle \sum _{i=1}^m}{v}_{id}{\tilde{x}}_{ij}=1\hfill \\ u_{rd}\ge 0,r=1,\dots ,s,\hfill \\ v_{id}\ge 0,i=1,\dots ,m.\hfill \end{array}$$

(3)

$$\begin{array}{c}{E}_{dj}^U=\max {\displaystyle \sum _{r=1}^s}{u}_{rd}{\tilde{y}}_{rj}\hfill \\ \mathrm{s}.\mathrm{t}.\hfill \\ {\displaystyle \sum _{r=1}^s}{u}_{rd}{\tilde{y}}_{rd}-E_{dd}{\displaystyle \sum _{i=1}^m}{v}_{id}{\tilde{x}}_{id}=0\hfill \\ {\displaystyle \sum _{r=1}^s}{u}_{rd}{\tilde{y}}_{rj}-{\displaystyle \sum _{i=1}^m}{v}_{id}{\tilde{x}}_{ij}\le 0,j=1,\dots ,n,j\ne d\hfill \\ {\displaystyle \sum _{i=1}^m}{v}_{id}{\tilde{x}}_{ij}=1\hfill \\ u_{rd}\ge 0,r=1,\dots ,s,\hfill \\ v_{id}\ge 0,i=1,\dots ,m.\hfill \end{array}$$

(4)

Model (3)—The Aggressive Model: This model is crafted with an aggressive strategy, focusing on minimizing a specific output. The essence of this model lies in its competitive drive, striving to reduce costs or mitigate negative impacts to the greatest extent feasible.

Model (4)—The Benevolent Model: In contrast, Model (4) embodies a benevolent strategy, with its objective function aimed at maximizing an output. This model is designed to optimize and amplify the benefits or positive outcomes to their fullest potential.

For DMU j ($j=1,2,\dots ,n$), the cross-efficiency evaluated by DMU d is bounded by the interval $[{\tilde{E}}_{dj}^L,{\tilde{E}}_{dj}^U]$. The generalized matrix of cross-efficiencies is detailed in

Table 2. Generalized matrix of cross-efficiency.

DMU	1	2	…	n
1	[${\tilde{E}}_{11}^L$, ${\tilde{E}}_{11}^U$ ]	[${\tilde{E}}_{12}^L$, ${\tilde{E}}_{12}^U$ ]	…	[${\tilde{E}}_{1n}^L$, ${\tilde{E}}_{1n}^U$ ]
2	[${\tilde{E}}_{21}^L$, ${\tilde{E}}_{21}^U$ ]	[${\tilde{E}}_{22}^L$, ${\tilde{E}}_{22}^U$ ]	…	[${\tilde{E}}_{2n}^L$, ${\tilde{E}}_{2n}^U$ ]
…	…	…	…	…
n	[${\tilde{E}}_{n1}^L$, ${\tilde{E}}_{n1}^U$ ]	[${\tilde{E}}_{n2}^L$, ${\tilde{E}}_{n2}^U$ ]	…	[${\tilde{E}}_{nn}^L$, ${\tilde{E}}_{nn}^U$ ]
Average	[${\overline{E}}_1^L,{\overline{E}}_1^U$]	[${\overline{E}}_2^L,{\overline{E}}_2^U$]	…	[${\overline{E}}_n^L,{\overline{E}}_n^U]$

.

Utilizing Gibbs’ entropy theory, ref. [37] introduced an entropy index derived from the cross-efficiency matrix. The entropy for the cross-efficiency of DMU j is defined as $H_j$, where:

$$H_j=-K_j\sum _{d=1}^n{G}_{dj}ln{G}_{dj}$$

(5)

where $K_j={\displaystyle \frac{{\overline{E}}_j^L+{\overline{E}}_j^U}{2}}$ and $G_{dj}={\displaystyle \frac{E_{dj}}{{\sum }_{d=1}^n{E}_{dj}}}$.

Given that the true cross-efficiency ${\widehat{E}}_{dj}$ for DMU j lies within the interval $[E_{dj}^L,E_{dj}^U]$, the model presented in Equation (5) can be expressed as:

$$H_j=-K_j\sum _{d=1}^n\left({\displaystyle \frac{{\widehat{E}}_{dj}}{{\sum }_{d=1}^n{\widehat{E}}_{dj}}}ln{\displaystyle \frac{{\widehat{E}}_{dj}}{{\sum }_{d=1}^n{\widehat{E}}_{dj}}}\right)$$

(6)

To identify the DMU that preserves the most information, we seek to maximize the entropy value as expressed in Equation (8). This objective is achieved through the following model:

$$\begin{array}{cc}\hfill {\widehat{H}}_j& =max\left(-K_j\sum _{d=1}^n\left({\displaystyle \frac{{\widehat{E}}_{dj}}{{\sum }_{d=1}^n{\widehat{E}}_{dj}}}ln{\displaystyle \frac{{\widehat{E}}_{dj}}{{\sum }_{d=1}^n{\widehat{E}}_{dj}}}\right)\right)\hfill \\ \hfill & \mathrm{s}.\mathrm{t}.E_{dj}^L\le {\widehat{E}}_{dj}\le E_{dj}^U,d=1,\dots ,n.\hfill \end{array}$$

(7)

In Equation (7), ${\widehat{H}}_j$ denotes the maximum entropy value for DMU j. The goal is to solve for the set ${\widehat{E}}_{dj}$ for $d=1,2,\dots ,n$ that achieves this maximum entropy. The model’s objective function is nonlinear and fractional, presenting bound constraints. Drawing on the approach by [20], we introduce $t={\displaystyle \frac{1}{{\sum }_{d=1}^n{\widehat{E}}_{dj}}}$ and $w_{dj}=t{\widehat{E}}_{dj}$, therefore transforming the model into a linearly constrained form as shown in Equation (8).

$$\begin{array}{c}{\widehat{H}}_j=\max \left(-K_j\sum _{d=1}^n{w}_{dj}ln{w}_{dj}\right)\hfill \\ \mathrm{s}.\mathrm{t}.\sum _{d=1}^n{w}_{dj}=1\hfill \\ t{E}_{dj}^L\le w_{dj}\le t{E}_{dj}^U\hfill \\ t>0,d=1,\dots ,n.\hfill \end{array}$$

(8)

Equation (8) aims to maximize a concave function within the bounds of linear constraints, ensuring the identification of a global optimum. Solving this model provides the optimal values for ${\widehat{H}}_j^{*}$, $w_{dj}^{*}$, and $t^{*}$. Utilizing the relationship $w_{dj}=t{\widehat{E}}_{dj}$, we derive the optimal ${\widehat{E}}_{dj}^{*}$, aligning with the maximum entropy. Therefore, the optimal cross-efficiency value for DMU j ($j=1,2,\dots ,n$) is determined as:

$${\overline{E}}_j^{*}={\displaystyle \frac{1}{n}}\sum _{d=1}^n{\widehat{E}}_{dj}^{*}$$

(9)

Having obtained the optimal cross-efficiency value ${\tilde{E}}_j^{*}$ that minimizes entropy, we proceed to substitute the CCR efficiency ${\theta }_j$ with ${\overline{E}}_j^{*}$ as suggested by [11], yielding the following result:

$$\begin{array}{c}{\tilde{E}}_j^{*}={\displaystyle \sum _{r\in Q^{\prime }}}{\beta }_r{\tilde{y}}_{rj}-{\displaystyle \sum _{i\in P^{\prime }}}{\alpha }_i{\tilde{x}}_{ij}+{\tilde{\epsilon }}_j,j=1,2,\dots ,N\hfill \\ {\alpha }_i,{\beta }_r\ge 0,\forall i\in P^{\prime },r\in Q^{\prime }.\hfill \end{array}$$

(10)

where ${\alpha }_i$ and ${\beta }_r$ denote the coefficients for the i-th input and r-th output, respectively. The error terms ${\tilde{\epsilon }}_j$ are independently and identically distributed as a normal distribution with mean 0 and variance ${\sigma }_{\tilde{\epsilon }}^2$. P, Q represent the set of input and output variables, where $P^{\prime },Q^{\prime }$ are nonempty subsets, with $P^{\prime }$ being a subset of P and $Q^{\prime }$ being a subset of Q ($P^{\prime }\subseteq P,Q^{\prime }\subseteq Q$).

These stochastic problems pose theoretical challenges, making the Monte Carlo simulation an essential tool for efficiency estimation under stochastic conditions. However, determining the optimal number of simulation replications remains an open question [11]. In a static model, selecting a inputs and b outputs requires $N_{ab}$ computational and estimation runs to explore all possible AIC values.

$$N_{ab}=\left({\displaystyle \genfrac{}{}{0pt}{}{m}{a}}\right)\times \left({\displaystyle \genfrac{}{}{0pt}{}{s}{b}}\right)$$

(11)

In the stochastic model, the challenge lies in the need for multiple simulations for each combination of input and output variables. Let $T_i$ represent the number of simulations for the i-th variable combination, with i ranging from 1 to $N_{ab}$. We denote $\tau$ as the total simulation budget. The variable-selection model in a stochastic environment can then be formulated as follows:

$$\begin{array}{c}\min AIC(P^{\prime },Q^{\prime },T_i)={\displaystyle \frac{1}{T_i}}{\displaystyle \sum _{l=1}^{T_i}}\left(n\left[ln\left(2\pi \right)+ln{\widehat{\sigma }}_{{\epsilon }_l}^2+1\right]+2\left(\left|P^{\prime }\right|+\left|Q^{\prime }\right|\right)\right)\hfill \\ \mathrm{s}.\mathrm{t}.T_1+T_2+\dots +T_{N_{ab}}=\tau \hfill \end{array}$$

(12)

where

$${\widehat{\sigma }}_{{\epsilon }_l}^2={\displaystyle \frac{1}{n}}\sum _{j=1}^n{\left({\tilde{E}}_j^{*}+\sum _{i\in P^{\prime }}{\alpha }_i{\tilde{x}}_{ij}-\sum _{r\in Q^{\prime }}{\beta }_r{\tilde{y}}_{rj}\right)}^2$$

(13)

3.2. The Model of OCBA

Optimal Computing Budget Allocation (OCBA) is a strategic simulation optimization approach that intelligently allocates additional simulations to variable combinations with the potential to yield lower Akaike Information Criterion (AIC) values [16]. This method is crucial for identifying the optimal number of simulation replications and selecting the variable combination that minimizes the AIC value. In the application of the OCBA model, we introduce the following notation:

• $\gamma$: Total number of simulation replications.
• ${\overline{B}}_i$: Sample mean of the AIC for the i-th variable combination.
• ${\sigma }_i^2$: Variance of the AIC for the i-th variable combination.
• $c_1$: The variable combination with the smallest estimated AIC.
• $c_2$: The variable combination with the second smallest estimated AIC.
• $T_i$: Number of replications allocated to the i-th variable combination.
• ${\delta }_{i,j}$: The difference between the sample means of the AIC for combinations i and j, calculated as ${\delta }_{i,j}={\overline{B}}_i-{\overline{B}}_j$.

These parameters facilitate the determination of simulation runs for each design, ensuring that computational effort is directed toward the most promising variable combinations.

$${\displaystyle \frac{T_{c_1}}{T_{c_2}}}={\displaystyle \frac{{\sigma }_{c_1}}{{\sigma }_{c_2}}}\sqrt{\sum _{j=1}^l{\left({\displaystyle \frac{{\delta }_{c_1,c_2}}{{\delta }_{c_1,i}}}\right)}^2},i\ne c_1$$

(14)

$${\displaystyle \frac{T_{c_1}}{T_{c_2}}}={\left({\displaystyle \frac{{\sigma }_{c_1}/{\delta }_{c_1,i}}{{\sigma }_{c_2}/{\delta }_{c_1,c_2}}}\right)}^2,i\ne c_1,c_2$$

(15)

At the outset, we initiate the process by simulating all l combinations with $t_o$ replications each. As the simulations progress, the sample means ${\overline{B}}_i$ and variances ${\sigma }_i^2$ for each combination are iteratively updated. The simulation budget is then incremented by d, and Equations (14) and (15) are utilized to determine the number of new simulations required. This process continues until the total simulation budget is fully utilized. The OCBA allocation procedure is outlined as follows:

(1)

Initialize: Set $t=1$ and conduct $t_o$ simulations for all l combinations. Initialize $T_1^t,T_2^t,\dots ,T_l^t$ to $t_o$.

(2)

Update: Calculate the sample means ${\overline{B}}_i$ and variances ${\sigma }_i^2$ for each combination.

(3)

Allocate: Increase the simulation budget by d and compute the new simulation counts $T_1^{t+1},T_2^{t+1},\dots ,T_l^{t+1}$ using Equations (14) and (15).

(4)

Simulate: For each combination i, perform an additional $T_i^{t+1}-T_i^t$ simulations.

(5)

Termination: If the cumulative simulation count ${\sum }_{i=1}^l{T}_i^t$ is less than the total budget $\gamma$, increment t by 1 and return to the Update step; otherwise, terminate the process.

4. Numerical Example with Stochastic Data

In this section, we validate our proposed method through a numerical example using a stochastic dataset from [11]. The dataset introduces random variations in inputs and outputs as follows: ${\tilde{X}}_{ij}^d=X_{ij}^d+{\widehat{\epsilon }}_j^{dx}$ and ${\tilde{Y}}_{rj}^d=Y_{rj}^d+{\widehat{\epsilon }}_j^{dy}$, where ${\widehat{\epsilon }}_j^{dx}$ and ${\widehat{\epsilon }}_j^{dy}$ represent error terms. These error terms are normally distributed with zero mean and variances of 0.36 and 0.6, respectively (${\widehat{\epsilon }}_j^{dx}\sim N(0,0.36)$; ${\widehat{\epsilon }}_j^{dy}\sim N(0,0.6)$). The deterministic components of the variables are detailed in

Table 3. Data set from [11].

DMU	X1	X2	Y1	Y2	Y3
1	4	3	2	5	3
2	8	5	3	4	1
3	5	7	5	9	4
4	2	8	1	3	2
5	1	3	6	7	1
6	4	1	2	6	4

.

Given the limited literature on stochastic variable selection, we selected two studies for comparison: [11,27]. As shown in

Table 4. Comparison of Methods.

No.	X1	X2	Y1	Y2	Y3	Our Method		Ref. [11]		Ref. [27]
						Reps.	AIC1	Reps.	AIC2	Imp.	Reps	AIC3	Imp.
1	1	0	1	0	0	466	−3.94	100	−1.98	99%	100	−2.38	65%
2	1	0	1	1	0	157	−0.98	100	0.66	248%	100	1.16	184%
3	0	1	0	0	1	97	−0.91	100	1.24	173%	100	2.44	137%
4	1	0	0	1	0	10	2.11	100	3.11	32%	100	4.28	51%
5	0	1	0	1	1	34	2.02	100	3.95	49%	100	5.59	64%
6	0	1	0	1	0	10	2.75	100	4.34	37%	100	5.61	51%
7	1	1	0	1	0	10	4.38	100	5.05	13%	100	6.05	28%
8	0	1	1	0	1	10	3.38	100	5.50	39%	100	7.07	52%
9	0	1	1	1	0	12	3.85	100	5.62	31%	100	6.83	44%
10	1	1	1	1	0	10	6.34	100	7.12	11%	100	8.55	26%
11	0	1	1	0	0	32	3.93	100	5.65	30%	100	7.27	46%
12	0	1	1	1	1	10	5.22	100	6.89	24%	100	8.31	37%
13	1	0	1	0	1	31	4.44	100	5.91	25%	100	7.21	38%
14	1	0	0	1	1	10	4.56	100	6.09	25%	100	7.62	40%
15	1	0	0	0	1	10	5.41	100	5.88	8%	100	7.14	24%
16	1	1	1	0	0	10	5.54	100	7.26	24%	100	9.50	42%
17	1	0	1	1	1	19	5.70	100	7.32	22%	100	9.35	39%
18	1	1	0	1	1	10	6.62	100	7.99	17%	100	10.53	37%
19	1	1	1	0	1	10	7.34	100	8.23	11%	100	10.38	29%
20	1	1	0	0	1	12	6.31	100	7.04	10%	100	9.10	31%
21	1	1	1	1	1	10	8.12	100	9.66	16%	100	10.30	21%
Total						980		2100			2100
Average						47		100		45%			52%

Note: the improvement (imp.) of AIC is calculated by ${\displaystyle \frac{|AI{C}_1-AI{C}_2|}{|AI{C}_2|}}\times 100\%$ and ${\displaystyle \frac{|AI{C}_1-AI{C}_3|}{|AI{C}_3|}}\times 100\%$.

, these studies use a fixed number of simulations for AIC evaluation, unlike our OCBA–AIC method that intelligently allocates computational resources. This dynamic allocation in our approach leads to a more efficient variable-selection process, as demonstrated by the lower AIC values achieved, indicating a superior fit to the data.

Table 4 presents a comparative analysis of the optimal combinations of inputs and outputs under varying numbers of variables, utilizing two distinct methods. The left portion of the table details the results obtained through our Optimal Computing Budget Allocation (OCBA) method, while the right part displays the outcomes from the model presented in [11,27]. All methods converge on the identification of the core service model ($|P^{\prime }|=|Q^{\prime }|=1$), where input variable X1 and output variable Y1 emerge as the optimal selection from the possible 21 variable combinations.

Notably, our OCBA approach, despite yielding the same variable set as the methods in [11,27], consistently delivers a lower AIC value. This achievement is attributed to the enhancement of the original AIC model with an optimal cross-efficiency estimator, which strategically allocates increased simulation replications to variable combinations with the potential to yield even lower AIC scores.

Our comprehensive reanalysis has demonstrated that with a mere 980 simulation replications, our method not only matches but exceeds the performance of those requiring 2100 replications. Remarkably, our OCBA-enhanced approach has achieved an average reduction of 45% and 52% in AIC scores compared to the traditional approaches. This significant enhancement underscores the efficiency, robustness, and computational thrift of our method while preserving or enhancing the quality of the outcomes. The integration of OCBA with DEA represents a notable advancement in the field, offering a more efficient and effective strategy for simulation-based optimization.

In this paper, we extend the existing variable-selection methods in stochastic environments by introducing two key innovations: First, we enhance efficiency estimation by incorporating both self and peer evaluations using a game cross model. Second, we address the challenge of determining the optimal number of simulation replications in stochastic variable selection by integrating the Akaike Information Criterion (AIC) with the Optimal Computing Budget Allocation (OCBA) technique. This approach provides a compelling case for the adoption of our method in relevant applications, offering a superior performance in terms of statistical rigor and computational efficiency.

5. Real Case Study

5.1. Supplier Selection

Our proposed model offers a structured framework for variable selection within DEA, applicable across various implementation scenarios. Specifically, this section addresses a supplier selection challenge, drawing on insights from [28]. The associated input and output variables are outlined in

Table 5. Data from an experimental study reported in [28].

	X1	X2	X3	X4	Y1	Y2	Y3
DMU1	(152;153;229)	(280;282;421)	299	(272;274;410)	(13.0;14.5;15.9)	(98.1;109.0;120.0)	(8.5;9.5;10.4)
DMU2	(118;118;132)	(242;242;272)	160	(235;235;264)	(9.4;10.5;11.5)	(66.6;74.0;81.4)	(5.6;6.2;6.8)
DMU3	(109;109;122)	(245;245;270)	205	(238;238;268)	(9.0;10.0;11.0)	(67.6;75.1;82.6)	(7.9;8.8;9.6)
DMU4	(34;34;39)	(93;93;107)	120	(101;101;128)	(4.7;5.2;5.8)	(31.2;34.7;38.1)	(2.1;2.3;2.6)
DMU5	(92;106;123)	(176;203;237)	276	(180;207;242)	(7.0;7.8;8.6)	(51.3;57.0;62.7)	(9.8;10.9;12.0)
DMU6	(143;157;182)	(198;217;252)	233	(251;275;318)	(13.0;14.5;15.9)	(85.9;95.4;105.0)	(10.7;11.9;13.1)
DMU7	(122;122;194)	(215;215;339)	257	(157;157;248)	(8.3;9.2;10.2)	(60.39;67.1;73.8)	(11.2;12.5;13.7)
DMU8	(162;162;249)	(338;338;518)	635	(169;169;259)	(6.9;7.7;8.5)	(55.1;61.3;67.4)	(9.7;10.7;11.8)
DMU9	(185;185;304)	(376;376;625)	481	(336;336;557)	(18.7;20.7;22.8)	(145.8;162.0;178.2)	(23.4;25.9;28.5)
DMU10	(106;106;134)	(296;296;379)	339	(173;173;235)	(7.3;8.2;9.0)	(53.4;59.4;65.3)	(5.2;5.7;6.3)
DMU11	(933;933;1033)	(1782;1782;1955)	1762	(1392;1392;1532)	(48.9;54.4;59.8)	(366.5;407.2;447.9)	(94.6;105.1;115.6)
DMU12	(69;86;71)	(124;155;129)	123	(104;130;130)	(4.4;4.9;5.4)	(19.2;21.4;23.5)	(8.3;9.3;10.2)
DMU13	(57;68;57)	(94;113;94)	123	(101;122;122)	(3.7;4.1;4.5)	(26.8;29.8;32.7)	(2.4;2.7;3.0)
DMU14	(34;35;35)	(88;90;89)	150	(226;230;230)	(2.8;3.2;3.4)	(69.0;76.7;84.4)	(0.0;0.0;0.0)
DMU15	(240;240;243)	(668;668;672)	439	(571;571;600)	(21.8;24.3;26.7)	(155.9;173.2;190.5)	(37.3;41.4;45.6)

.

In this dataset, with $m=4$ inputs, $s=3$ outputs, and $n=15$ decision-making units (DMUs), we apply a heuristic that the sum of selected inputs and outputs, $|P^{\prime }|+|Q^{\prime }|$, should not exceed 5. Consequently, our method evaluates all possible variable combinations to identify the optimal set, ensuring the total number of variables does not surpass this limit. The outcomes are detailed in

Table 6. Optimal combination for different number of variables.

No.	X1	X2	X3	X4	Y1	Y2	Y3	Total Variables	Rep. Times	AIC
K1	1	0	0	0	0	1	0	2	410	0.3397
K2	1	1	0	0	0	1	0	3	510	3.6151
K3	1	0	0	0	0	1	1	3	517	15.4299
K4	1	1	1	0	0	1	0	4	662	14.8139
K5	1	0	0	0	1	1	1	4	467	22.0476
K6	1	0	1	0	0	1	1	4	543	23.1462
K7	1	1	1	0	1	1	0	5	504	26.9565
K8	1	0	1	0	1	1	1	5	479	28.0382
K9	1	1	1	1	0	1	0	5	519	28.9712

. Notably, an increase in the number of selected variables correlates with a higher average AIC value, indicating greater information redundancy. As discussed in Section 2, a lower AIC signifies a more efficient model with less redundancy. Therefore, in DEA model variable selection, it is crucial to minimize the number of inputs and outputs while maintaining the necessary information to avoid redundancy.

When decision-makers aim to select 5 variables from a total of 7 inputs and outputs, our model guides them to identify the optimal combination by omitting 2 variables. With the constraint $|P^{\prime }|+|Q^{\prime }|=5$, the model suggests three potential configurations: 4 inputs and 1 output, 3 inputs and 2 outputs, or 2 inputs and 3 outputs. Table 6 reveals that the combination (X1, X2, X3, Y1, Y2) yields the minimum AIC value of 26.957, making it the optimal selection. This indicates that variables X4 and Y3 contribute least to performance measures and can be excluded with minimal impact on information retention.

For scenarios where decision-makers are interested in selecting 3 or 4 variables, our model provides alternative optimal combinations. With 4 variables, the combination (X1, X2, X3, Y2) is optimal with an AIC value of 14.814. When selecting only 3 variables, the combination (X1, X2, Y2) is favored, with the lowest AIC value of 3.615. Our method also aids in identifying core variables crucial for company operations by maintaining efficiency ranks close to the original model even after other variables are omitted. In Table 6, X1 and Y2 emerge as key variables, highlighting their significance in the DEA model.

Table 7. DMU rankings with the Borda method.

Supplier	K1	K2	K3	K4	K5	K6	K7	K8	K9	Borda Score	Ranking
DMU14	1	1	1	1	2	1	2	2	1	123	1
DMU4	2	3	5	8	1	9	1	1	8	97	2
DMU15	3	9	2	4	3	2	5	3	7	97	3
DMU6	7	2	9	3	7	4	4	5	3	91	4
DMU9	4	4	4	7	4	5	8	7	2	90	5
DMU2	6	8	11	2	10	3	3	4	4	84	6
DMU3	5	7	8	5	6	6	6	6	9	77	7
DMU1	8	6	12	6	11	8	7	8	5	64	8
DMU5	9	10	6	12	8	11	11	10	12	46	9
DMU12	14	14	3	14	5	7	10	9	15	44	10
DMU7	12	11	10	9	12	12	12	11	6	40	11
DMU13	10	5	13	10	13	13	9	12	13	37	12
DMU11	13	12	7	11	9	10	13	13	14	33	13
DMU10	11	13	14	13	14	14	14	14	10	18	14
DMU8	15	15	15	15	15	15	15	15	11	4	15

presents the ranking of decision-making units (DMUs) based on the Borda method, a voting system that assigns points to alternatives based on their order of preference. In this method, each DMU receives points for its ranking in each criterion, with the DMU ranking first receiving $n-1$ points, where n is the total number of DMUs. The points are then summed across all criteria to obtain the Borda score for each DMU. The DMU with the highest Borda score is considered to have the most robust performance. According to the findings, DMU14 emerges as the top-ranked DMU, achieving the highest Borda score of 123 points and securing the first rank. This indicates its consistent superior performance across various criteria. Conversely, DMU8 is identified as the lowest-ranked DMU, with a Borda score of 4 points, and is consistently placed at the bottom rank of 15 across all criteria. These results underscore that DMU14 demonstrates the most robust performance and should be the preferred choice for a supplier partnership, providing the decision-maker with a data-driven basis for selection. The Borda method ensures a comprehensive evaluation by considering the overall performance rather than focusing on a single criterion, thus facilitating a more informed decision-making process.

5.2. Tourism Technology Jobs Ranking

We have amassed data across 220 unique decision-making units (DMUs) within the tourism technology industry, with each DMU representing a distinct job profile. Our analytical framework meticulously assesses the attractiveness of these opportunities. The dataset includes key variables that shape job desirability, as detailed in

Table 8. Descriptive Statistics for Tourism Technology Company with 220 DMUs.

Variable	N	Minimum	Maximum	Mean	Standard Deviation
Experience (X1)	220	0.3	10.0	2.957	2.0802
Education (X2)	220	14.0	22.0	15.582	1.0103
Skill (X3)	220	15.0	45.0	29.527	5.3143
Environment (Y1)	220	2.0	20.0	7.891	5.2861
Welfare (Y2)	220	1.0	76.0	27.336	19.3373
Salary (Y3)	220	0.4	45.0	17.209	7.7670

. Here, input variables such as Experience (X1), Education (X2), and Skill (X3) represent the candidate’s qualifications and the prerequisites for the role. Conversely, output variables like Environment (Y1), Welfare (Y2), and Salary (Y3) reflect the incentives and work conditions provided by the companies, offering a comprehensive view of the value proposition for potential employees.

Experience (X1) reflects the depth of professional background expected for a position, Education (X2) indicates the level of academic achievement required, and Skill (X3) assesses the proficiency and expertise necessary for the role. Environment (Y1) relates to the work atmosphere and organizational culture, Welfare (Y2) covers the range of benefits beyond salary, and Salary (Y3) is the monetary value offered for the job. Utilizing our advanced methodological approach, we rank these tourism technology companies based on the composite attractiveness of their job offerings, providing insights into the job market’s trends and assisting in strategic decision-making for both employers and potential candidates.

Table 9. Optimal Combinations with Corresponding AIC Scores.

ID	X1	X2	X3	Y1	Y2	Y3	Total Variables	Rep. Times	AIC
K1	1	0	0	0	1	0	2	462	−2.291644
K2	1	1	0	0	1	0	3	391	−1.665309
K3	1	0	0	0	1	1	3	325	2.252313
K4	1	1	1	0	1	0	4	571	5.488801
K5	1	0	0	1	1	1	4	361	6.319803
K6	1	0	1	0	1	1	4	513	9.650136
K7	1	1	1	0	1	0	4	375	9.866833
K8	1	0	1	1	1	1	5	652	9.952996
K9	1	1	1	0	1	1	5	457	10.024476
K10	1	1	1	1	1	1	6	559	10.066365

presents the optimal combinations of input and output variables for each DMU, along with their corresponding AIC scores. These combinations are determined by maximizing the AIC scores, indicating the most efficient variable configurations for each DMU. The table provides a detailed breakdown of the input and output variables, the total number of variables, the number of replications used in the analysis, and the AIC scores. This ranking system allows for a comparative analysis of the job offerings, providing insights into the job market’s trends and assisting in strategic decision-making for both employers and potential candidates.

Furthermore,

Table 10. Ranking of Tourism Technology Jobs.

Job	K1	K2	K3	K4	K5	K6	K7	K8	K9	K10	Borda Score	Ranking
DMU92	1	2	5	1	9	12	1	5	3	6	2155	1
DMU21	2	4	16	11	6	7	2	3	1	2	2146	2
DMU199	18	10	15	3	1	4	4	10	6	5	2124	3
DMU219	5	1	1	9	21	3	13	12	8	11	2116	4
DMU27	4	5	2	4	3	8	17	23	4	18	2112	5
DMU74	6	16	7	12	12	1	15	2	5	12	2112	6
DMU169	13	9	6	5	2	9	3	7	25	13	2108	7
DMU148	14	19	12	16	13	5	8	1	11	1	2100	8
DMU197	9	15	11	17	7	2	14	14	9	4	2098	9
DMU146	8	6	8	22	5	18	18	9	7	8	2091	10
…	…	…	…	…	…	…	…	…	…	…	…	…
DMU38	209	205	201	210	208	218	203	217	208	217	104	211
DMU163	207	216	200	216	212	212	208	212	215	207	95	212
DMU126	212	208	205	217	214	209	210	213	209	210	93	213
DMU150	213	214	211	209	215	197	215	208	218	209	91	214
DMU184	205	211	213	215	200	216	198	219	213	219	91	215
DMU175	217	209	218	218	199	215	209	206	212	212	85	216
DMU30	218	218	212	212	213	205	216	210	207	205	84	217
DMU73	219	217	214	206	211	200	217	216	203	216	81	218
DMU10	215	220	217	220	219	220	220	218	220	220	11	219
DMU35	220	219	220	219	220	219	219	220	217	218	9	220

enhances our analysis by ranking tourism technology jobs using a Borda score, which integrates various criteria to measure job appeal. This ranking system, distilled from the optimal combinations in Table 9, offers a nuanced view of the job market’s most attractive and least compelling opportunities. It serves as a valuable guide for strategic decision-making in the tourism technology sector, aiding both employers in refining their recruitment strategies and job seekers in making informed career choices. Due to space constraints, the table highlights the top and bottom ten jobs, focusing on those that are particularly sought-after and those that may require improvements to enhance their appeal. The Borda score, a reliable statistical measure, ensures a comprehensive evaluation, providing a solid benchmark for assessing job attractiveness based on a holistic set of factors, including qualifications, benefits, and market trends. This ranking not only identifies the top ten most enticing job prospects but also reveals the ten that are least favored, offering a vivid portrayal of the job market’s standout and underperforming positions.

Based on the data presented in Table 10, DMU92 emerges as the most sought-after job type, securing the top position with a Borda score of 2155 points. This ranking indicates that DMU92 provides a compelling package of qualifications and benefits that resonate strongly with prospective candidates. On the other end of the spectrum, DMU35 is identified as the least popular, with a Borda score of 9 points, suggesting that it may lack the competitive edge in the job market, possibly due to an unappealing mix of qualifications and benefits on offer. These insights from the Borda score analysis offer valuable guidance for companies to refine their job offerings and for job seekers to identify the most attractive opportunities in the tourism technology sector.

The AIC scores play a crucial role in this analysis, as they provide a quantitative measure of the attractiveness of each job offering. A lower AIC score indicates a better fit of the model to the data, which in this context means a more attractive job profile. The AIC scores are used to rank the job offerings, allowing companies to assess the effectiveness of their recruitment strategies and job seekers to make informed decisions about which jobs to pursue.

By comparing the AIC scores, both companies and candidates can identify trends and make strategic decisions. Companies can adjust their job postings to better align with the qualifications and benefits that are most valued by the job market, while candidates can focus their job search on positions that offer the best combination of qualifications, prerequisites, and incentives. This analysis contributes to a more efficient and effective recruitment process in the tourism technology industry.

Our advanced methodological approach not only ranks the tourism technology companies based on the composite attractiveness of their job offerings but also offers a data-driven perspective on the job market trends. This comprehensive analysis assists in strategic decision-making for both employers and potential candidates, ensuring that job seekers can make informed choices about their employment opportunities and employers can align their job profiles with the qualifications and expectations of potential candidates.

The strategic role of decision-making in aligning organizational goals with resource management is well-established, with a focus on optimizing the allocation of resources to achieve the best outcomes [15]. This study aims to contribute to the literature by leveraging the Optimal Computing Budget Allocation (OCBA) technique, which provides a systematic approach to allocating computational resources efficiently [16]. The application of OCBA in the context of Data Envelopment Analysis (DEA) allows for the determination of the optimal number of simulation replications and the selection of the best combination of input and output variables. This approach is particularly beneficial in stochastic environments where traditional methods may fall short in capturing the inherent variability and uncertainty [13].

By integrating advanced analytics and statistical techniques such as AIC, this study enhances the robustness of DEA models, leading to more accurate and reliable efficiency measurements [12]. The use of AIC in model selection ensures that the most appropriate model is chosen based on its goodness of fit and simplicity, therefore improving the decision-making process [13]. Furthermore, the insights from career counseling and job market dynamics provide a foundation for understanding the multifaceted nature of decision-making, emphasizing the importance of aligning individual and organizational goals.

6. Conclusions

In this study, we introduce a novel method, OCBA–AIC, designed to select the optimal combinations of inputs and outputs in stochastic DEA. Our approach enhances existing variable-selection techniques by incorporating the stochastic nature of the variables. Specifically, we build upon the game cross-efficiency DEA model by leveraging the OCBA technique and Monte Carlo simulations.

Our method effectively integrates traditional and cross-DEA models, considering both self and peer evaluations to identify the most relevant variables based on efficiency entropy values and the Akaike Information Criterion (AIC). By employing the OCBA technique, OCBA–AIC not only determines the optimal number of simulation replications but also identifies the best combinations of inputs and outputs. Additionally, this method provides decision-making units (DMUs) with insights into which variables have the most significant impact on maintaining efficiency.

Our empirical results both corroborate and diverge from previous studies, highlighting the importance of context in variable selection and efficiency measurement. For instance, while our findings align with Nataraja et al. (2011) regarding the necessity for tailored variable selection, discrepancies arise in certain sectors where traditional assumptions of linearity may not hold [27]. From a practical perspective, our results have significant implications for organizations seeking to enhance efficiency measurement methodologies. By clearly identifying the most impactful variables, organizations can make more informed decisions regarding resource allocation and performance evaluation, ultimately driving operational improvements [38].

The implications of this research are significant for decision-making units operating in dynamic and uncertain environments. The ability to make informed decisions based on a comprehensive evaluation of variables and their interactions is crucial for strategic planning and resource allocation [14]. This study, therefore, not only advances the theoretical understanding of DEA in stochastic settings but also offers practical benefits to decision-makers, equipping them with the tools to navigate complexity and uncertainty in their decision-making processes.

Furthermore, our study offers valuable insights into the identification of a core efficiency model utilizing a single input and output. Overall, this method serves as a robust decision-making tool for DEA variable selection in stochastic environments, particularly due to its capability to rank DMUs effectively.

It is crucial to recognize the limitations inherent in this study. The efficacy of the OCBA–AIC method is heavily dependent on the quality and accuracy of the data, which may be subject to errors or biases [39]. Additionally, the method’s reliance on stochastic assumptions could limit its applicability in contexts where these assumptions do not hold or where data are limited. The computational intensity of the method may also pose challenges for less powerful systems. To address these limitations, future research should integrate advanced analytical techniques, such as machine learning and fuzzy logic, to refine variable selection and strengthen the robustness of DEA models [40]. Efforts should also be directed towards exploring alternative data sources, validating the method in various contexts, and developing more efficient computational algorithms.

In conclusion, the OCBA–AIC method introduced in this study provides a comprehensive framework for variable selection in DEA, offering both theoretical advancements and practical applications. It enhances the field’s ability to address the complexities of stochastic environments and provides a foundation for future research and development in performance management and decision-making.