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The aim of this study was to assess the relative efficiency of the Iranian forest management plans using the nonparametric method – Data Envelopment Analysis (DEA) as a well-known and robust technique for measuring the relative efficiency of organizations. The relative efficiency of forest management plans was calculated using the most frequency DEA models such as global technical efficiency (CCR), local pure technical efficiency (BCC) and Scale Efficiency (SE) on 12 units in Guilan Province, Iran. According to the results of CCR and BCC models, the efficiency averaged 0.83 and 0.93, respectively. The results of SE discussed a worrying aspect of these units efficiency; namely, there were only 3 efficient forest management plans (Shafaroud, Nav and Fiyab). However, the Scale Efficiency Index (SEI) brings out some interesting points; there were approximately 58% (7 units out of 12) under Increasing Returns to Scale (IRS). Therefore, the managers of forest management plans should focus more on the plans under IRS, so that they will have the opportunity to become more efficient through growth, otherwise managers will not be able to promote their overall productivity.

The aim of this research was to measure the relative efficiency of forest management plans in north of Iran. In order to fulfill the research, data of 12 forest management plans were collected from the financial balance sheets of Shafaroud Forest Company during a ten years period. First of all, basic Data Envelopment Analysis (DEA) models (BCC and CCR) were used to determine the efficiency. Then, due to the structure of the forest management plan, cost efficiency and revenue efficiency models based on DEA were used in order to measure the efficiency. Results indicated that 8 forest management plans were efficient based on BCC and CCR models. Furthermore, the results indicated that only one forest management plan was efficient based on cost efficiency and revenue efficiency models. These results could be due to the input oriented properties of the models, rational management and optimal use of resources.

Abstract In the data envelopment analysis (DEA) literature, productivity change captured by the Malmquist productivity index, especially in terms of a deterministic environment and stochastic variability in inputs and outputs, has been somewhat ignored. Therefore, this study developed a firm-specific, DEA-based Malmquist index model to examine the efficiency and productivity change of banks in a stochastic environment. First, in order to estimate bank-specific efficiency, we employed a two-stage double bootstrap DEA procedure. Specifically, in the first stage, the technical efficiency scores of banks were calculated by the classic DEA model, while in the second stage, the double bootstrap DEA model was applied to determine the effect of the contextual variables on bank efficiency. Second, we applied a two-stage procedure for measuring productivity change in which the first stage included the estimation of stochastic technical efficiency and the second stage included the regression of the estimated efficiency scores on a set of explanatory variables that influence relative performance. Finally, an empirical investigation of the Iranian banking sector, consisting of 120 bank-year observations of 15 banks from 2014 to 2021, was performed to measure their efficiency and productivity change. Based on the findings, the explanatory variables (i.e., the nonperforming loan ratio and the number of branches) indicated an inverse relationship with stochastic technical efficiency and productivity change. The implication of the findings is that, in order to improve the efficiency and productivity of banks, it is important to optimize these factors.

Since the healthcare system is one of the most important pillars of community health, and considering that providing healthcare services to the people is one of the elements of individual development in any country, attention and supervision of this sector can lead to development and social welfare. To ensure better and higher quality healthcare services, performance evaluation in the health sector plays a crucial role. In order to achieve this, proper and proportional use of existing facilities and assets is inevitable. In this study, by introducing an application in the field of healthcare systems, the educational hospitals of the country have been measured in terms of performance and their managerial ability has been calculated. Additionally, by identifying and introducing the impact of contextual variables on the performance of decision-making units, their efficiency has been assessed. For this purpose, data related to educational hospitals in 31 provinces of the country was collected, and then by identifying contextual variables and with the presence of undesirable factors, the efficiency was evaluated and the managerial ability of each was calculated. To reach this goal, in the first step, technical efficiency with the presence of undesirable factors was calculated using data envelopment analysis technique, and then the logarithm of technical efficiency obtained from the first stage was regressed on a set of contextual variables that affect hospital performance. In the next stage, managerial ability was extracted from the residual of the regression obtained from the previous stage. Finally, a unique ranking based on the managerial ability of each unit was provided. Ultimately, the results obtained were analyzed and examined in order to provide valuable suggestions for managers and more efficient management of the country's hospitals to maintain public health. According to the study, without considering contextual variables, 25 effective units were evaluated, but by applying the effect of contextual variables on the efficiency index, no unit becomes effective, proving the high impact of such indices on the performance of units. Additionally, in the ranking of units based on managerial ability, Lorestan province ranked first and Golestan province ranked last.IntroductionThe issue of increasing productivity and efficiency in healthcare costs is important for all countries. The health sector, by identifying the factors that affect community health precisely, influences national macroeconomic planning and minimizes their adverse effects on health. By utilizing the best practices in healthcare, significant improvements in the health of individuals and communities can be achieved. Therefore, proper investment in healthcare facilities and health centers, as well as improving the quality and efficiency of their services, is essential for sustainable development. In order to increase efficiency and productivity, understanding the current status and measuring the performance of hospitals in the healthcare system is of paramount importance. Ensuring the provision of better and higher quality health services requires evaluating the performance of the healthcare system. Therefore, it seems that employing efficiency measurement techniques and improving performance and productivity in this sector can improve processes and optimize the use of resources and the fair distribution of resources for the provision of desirable services. In recent years, various studies and methods have been proposed by researchers to measure the efficiency of decision-making units, which can be divided into two categories: parametric and non-parametric methods. Farrell (1957) first introduced the non-parametric method, and then Charnes et al. (1978) extended the initial analysis by Farrell from multi-input and single-output to multi-input and multi-output. The model developed by them was named the Charnes-Cooper-Rhodes model. Then, Banker et al. (1984) introduced the model. The non-parametric method is a linear programming-based method in which a linear programming problem is solved for each decision unit. This branch of operations research has rapidly advanced and is called data envelopment analysis. Data envelopment analysis is a mathematical programming technique for evaluating decision-making units and plays a fundamental role in identifying efficient boundaries and measuring the relative efficiency of units under scrutiny. Data envelopment analysis allows for the comparison of units with each other. Considering the importance of the health sector in improving the quality of life for individuals in society, we felt it necessary to examine the performance level and calculate the managerial capacity of hospitals in all 31 provinces of the country to ensure the proper functioning of this sector and take even small steps towards improving the quality of this sector. The aim of this research is to analyze and evaluate the performance of health sector hospitals in Iran in the presence of contextual variables and provide a ranking method based on managerial capacity. For this purpose, data related to educational hospitals in all 31 provinces of the country were collected, and then, by identifying contextual variables and the presence of undesirable factors, an attempt was made to evaluate the efficiency and calculate the managerial capacity of each hospital unit. To achieve this goal, in the first step, technical efficiency with the presence of undesirable factors was calculated using data envelopment analysis technique, and then the logarithm of technical efficiency resulting from the first step was regressed on a set of contextual variables that affect hospital performance. In the next step, managerial ability was extracted from the residual of the regression from the previous step. Finally, a unique ranking based on the managerial ability of each hospital was presented.MethodologyIn this article, based on studies conducted by Demerjian et al. (2020) and Banker et al. (2020), we examine the performance analysis and managerial abilities of 31 hospitals in the country through a three-stage process. Firstly, considering the presence of undesirable outputs, the efficiency analysis of the units of interest is obtained using the efficiency model proposed by Kuosmanen (2005) with the (3) technology. Then, using the least squares method, the impact of each of the contextual variables in this study, including "asset base", "density", and "number of physicians", on the efficiency scores obtained from the first stage is regressed. Subsequently, managerial ability is obtained from the residuals of the previous least squares method. Finally, a unique ranking based on the managerial ability of each hospital is presented.ResultsIn this study, which was conducted on the performance of the health care in Iran, a new ranking based on managerial ability was provided for comparing units. Based on calculations performed on a number of hospitals in 31 provinces of the country without considering contextual variables, 25 efficient units were evaluated. However, by applying the effect of contextual variables on the efficiency index, no unit appears to be efficient, proving the significant impact of contextual variables on the performance of units. Furthermore, the relationship between contextual variables and efficiency index was determined. For example, an increase in the amount of the contextual variable "number of physicians" will lead to an increase in managerial ability. This means that an increase in the number of physicians will benefit the improvement of the system's efficiency and managerial ability.ConclusionWithout a doubt, studying and investing in the healthcare industry is one of the most profitable and best areas for investment. In this regard, government hospitals in each country are one of the main and most important components of the healthcare sector. The hospitals studied in this research are considered as 3 government hospitals per province. Based on past efficiency studies, we find that each decision-making unit had its own specific inputs and outputs. The aim of this study is to analyze and examine the managerial ability of public hospitals in Iran. In this study, the performance of selected hospital units is analyzed in terms of managerial efficiency, considering the impact of other variables known as contextual variables on the performance of a decision-making unit. In this study, the performance of government hospitals in Iran is analyzed from a managerial perspective. The first step involves calculating the efficiency of units using basic models and considering undesirable outputs. Then, in the second step, the logarithm of technical efficiency obtained from the first step is regressed on a set of contextual variables that affect hospital performance. Furthermore, the impact of contextual variables, including total assets, physician density, and number of physicians, on the size of unit efficiency is measured in this study. Based on the results, 25 efficient units were evaluated, but with the application of contextual variables on efficiency indicators, no unit becomes efficient, proving the high impact of such indicators on unit performance. Additionally, based on the calculations performed, in the ranking of units with a managerial approach, Lorestan province ranks first and Golestan province, which has the weakest performance among the units under study, ranks last. The impact of contextual variables on efficiency indicators has been examined. For example, the impact of the "number of physicians" indicator on efficiency is direct, and a one-unit increase in it will lead to an increase in managerial efficiency. This means that an increase in the number of physicians will benefit the system's efficiency and managerial ability. However, the impact of the density variable, unlike the number of physicians, has an inverse effect on managerial ability. To provide suggestions for future studies, one can refer to generalizing the problem to the uncertainty space and studying different applications by bringing the problem into random spaces, providing more predictive predictions. Furthermore, this study can be implemented in analyzing performance and calculating managerial ability in various industries such as power plants, insurance industry, banks, etc., and based on the applications and the type of technology used, different approaches can be provided for calculating managerial

Abstract In performance analysis with tools such as data envelopment analysis, calculations of scale properties of the frontier points are studied using both qualitative and quantitative approaches. When the production process is a bit complicated, the calculation needs to be modified. Most existing studies are focused on a single-stage production process under the constant or variable returns to scale specification. However, some processes have two-stage structures, and, in such processes, the concepts of scale elasticity and returns to scale are inextricably related to the conditions of the stages of production. Thus, an evaluation of efficiency, scale elasticity, and returns to scale is sensitive to stages. In this study, we introduced a procedure to calculate technical efficiency and scale elasticity in a two-stage parallel-series production system. Then, our proposed technical efficiency and scale elasticity programs are applied to real data on 20 insurance companies in Iran. After applying our estimations to a real-world insurance industry, we found that, (i) overall, the total inputs of insurers in the life insurance sector should be reduced by 9%. Moreover, the inputs of nonlife insurers should be reduced by 50%. The final output in the investment sector must be increased by 48%. (ii) There are inefficiencies among all insurers in the investment sector, and to improve technical efficiency, the income from investments should be increased significantly. (iii) Finally, the efficiency and elasticity characterizations of insurers are directly subject to stages.

The electricity industry plays a pivotal role in a country's economic growth and development. Therefore, it is imperative to assess its performance and identify the strengths and weaknesses of its different sectors, such as production, transmission, and distribution, to enhance economic growth in diverse areas. Given the significance of the transmission sector, this research focuses on analyzing and evaluating the performance of 16 regional electricity companies in Iran from 1390 to 1398, with the aim of comprehending the impact of contextual variables on efficiency. To achieve this, the study will utilize two techniques - Data Envelopment Analysis (DEA) and Ordinary Least Squares (OLS) - to determine the efficiency score and estimate the effect of contextual variables on efficiency, respectively. In the first stage, the DEA technique is employed to calculate the technical efficiency of each company, considering their specific inputs and outputs. In the second stage, the logarithm of the efficiency scores obtained is regressed on contextual variables to establish their effect on efficiency. The residual derived from the regression is referred to as managerial ability. Finally, the companies are ranked based on their modified efficiency after removing the impact of contextual variables. Introduction The electricity industry comprises three key sectors: production, transmission, and distribution. It stands as one of the most crucial economic infrastructures in the country, exerting significant influence on industrial, agricultural, service, and other sectors. Undoubtedly, the growth of the electricity industry drives the nation's economic development and progress, contributing to the prosperity and comfort of its citizens (Tavassoli et al., 2020). Consequently, analyzing and examining the growth trajectory of each sector across different years becomes pivotal in mitigating adverse effects and fostering progress within this domain. In recent years, numerous researchers have conducted studies in this field. Some have independently evaluated each production, transmission, and distribution sector, while others have adopted a comprehensive approach by considering the integrated three-stage network structure. The research background highlights that the transmission sector has received less attention from researchers than other sectors. This is noteworthy because, following electricity production, the transmission process and energy accessibility to consumers are paramount. The absence of proper energy transfer can result in consumer dissatisfaction, financial losses, and stagnation within the competitive economic market. Therefore, identifying the strengths and weaknesses of the transmission sector's performance and comparing regional electricity transmission companies can effectively help enhance the performance level of each. One technique that has captured researchers' attention for evaluating the electricity industry's performance is the data envelopment analysis (DEA) technique. DEA is a non-parametric method used to assess the performance of homogeneous units, considering multiple inputs and outputs. It was initially introduced in 1978 by Charnes et al. The initial model was built upon the assumption of constant returns to scale. Subsequently, Banker et al. (1984) extended it by presenting a model under the assumption of returns to a variable scale. Importantly, traditional DEA models evaluate a system's performance based on specific inputs and outputs consumed and produced by the unit. However, various factors, such as contextual variables, managerial ability, and skill, can significantly influence performance and productivity. A crucial point to consider is that managerial abilities are not always overtly visible. This lack of direct visibility can impede accurate measurement. Hence, recognizing these variables among the existing indicators and assessing their influence on the performance and efficiency of each unit holds particular significance. This procedure enhances the precision of evaluation and opens avenues for delivering enhanced solutions aimed at improving the system's overall performance. Methodology The objective of this study is to analyze and evaluate the performance of Iran's regional electricity transmission sector while considering contextual variables and establishing a ranking methodology based on managerial ability. This perspective enables the identification of strengths and weaknesses in the system's structure from various angles and offers appropriate solutions for enhancement. To accomplish this, the first step involves identifying all variables within the transmission section, encompassing inputs, outputs, and contextual factors. Subsequently, we determine the technical efficiency of each regional power transmission company, taking into account specific inputs and outputs, using meta-frontier technology. The concept of meta-frontier in DEA measures the gap or distance between decision-making units (DMUs) across different boundaries. This approach assumes a unified boundary for all subgroups, enabling efficiency estimation based on a single boundary (Battese, 2004; O'Donnell, 2008). Its primary advantage lies in resolving the challenge of evaluating efficiency at varying levels. As a result, meta-frontier technology enhances the precision of evaluating regional power companies over multiple periods. After assessing the efficiency of each regional electricity transmission company, we employ the linear regression method to estimate the impact of contextual variables on efficiency, subsequently yielding a measure of managerial ability. Ultimately, we introduce a method for ranking each company based on managerial ability. The advantage of the proposed method is that, in addition to reviewing and analyzing the technical efficiency of each of the companies in the regional electricity transmission sector during different periods, it will be possible to evaluate the managerial ability of each of the companies. Such a perspective allows for companies to be compared from different dimensions. Moreover, providing a new ranking criterion based on managerial ability also facilitates a better and more accurate comparison. Results In this study, the performance of Iran's regional power companies was analyzed and evaluated from two systems and management perspectives during the years 1390-1398. Additionally, a new rating criterion based on managerial ability was presented to compare the performance of companies during 9 time periods. In this regard, firstly, the technical efficiency of 16 regional electricity companies during 9 time periods was calculated based on the inputs of the number of employees and receiving energy from neighboring companies and the outputs of sending energy to neighboring companies and delivering energy to distribution companies, using meta-frontier technology and the DEA approach. Then, the effect of contextual variables, such as line length, transformer capacity, and loss magnitude, on the efficiency score of each company was estimated using the ordinary least squares method (OLS). Furthermore, the managerial ability of each company was determined during different periods. Ultimately, a ranking criterion was established based on the results of technical efficiency after removing the effect of contextual variables. Conclusion The results of efficiency measurements over 9 time periods indicate that the highest and lowest average efficiencies were observed in the years 1390 and 1398, respectively. Furthermore, it's evident that, in general, the performance of Iran's 16 regional electricity companies exhibited a consistent upward trend from 1390 to 1398. Among the 16 evaluated companies, the Guilan regional electricity company consistently achieved the highest level of efficiency across all 9 time periods, reflecting its strong performance. Conversely, the Fars regional electricity company consistently had the lowest efficiency, indicating its weaker performance compared to other companies. When analyzing the companies' performance by year, it's noteworthy that the Tehran regional electricity company secured the highest rank in 1390, 1391, and 1394, while the Fars regional electricity company held the top spot in the remaining years. In contrast, the Sistan regional electricity company consistently displayed the lowest performance throughout all periods. The assessment of management performance over the 9 time periods indicates that the Kerman regional electricity company demonstrated superior performance from 1390 to 1393, whereas the Guilan regional electricity company excelled from 1394 to 1398, outperforming other companies. Conversely, the Gharb regional electricity company exhibited weaker performance compared to its counterparts. Additionally, the results of the regression analysis highlight a positive relationship between the efficiency score and two variables: line length and transformer capacity. Conversely, the relationship with loss magnitude is observed to be inversely correlated.

When the costs of the inputs and outputs of the units under evaluation are known, the evaluation of the profit efficiency of the units is one of the most significant evaluations that can provide valuable information about them. In this research, first, a new definition of the optimal scale size based on the maximization of the average measure of profit efficiency is presented. The average measure of profit efficiency develops the concept of economic efficiency measure by introducing a more accurate measure of efficiency compared to the measure of comparative and profit efficiency. It has been shown that the average measure of profit efficiency in a convex space is equivalent to the measure of profit efficiency in constant returns to scale technology, and then, some models are presented to calculate profit efficiency in a stochastic environment, to increase the ability of profit models in real examples by considering the calculation errors of inputs and outputs. Finally, the proposed method is used in an empirical example to calculate the average profit efficiency of a set of postal areas in Iran.

Congestion is one of the important concepts in data envelopment analysis that occurs when excessive inputs reduce the maximally possible outputs. Identification and elimination of congestion have a significant impact on reducing inputs along with increasing outputs. Hence, various studies have been conducted to detect and evaluate congestion. However, in today's world, no organization can achieve its final output with just one process of input. In other words, today's organizations have a network structure that consists of several subsections. Ignoring the existing influences among the subsections processes may lead to inadequate or even incorrect results for evaluating the congestion. While all of the existing methods only evaluate the congestion of each subsection or the whole system independently. Therefore, in this paper, the concept of congestion is developed for a specific and so practical case of network structure called “two-stage network structure”. This case of network structure consists of two series stage such that stage 1 consume some primary inputs to produce some intermediate outputs. In the following, the intermediate outputs are used as the inputs of stage 2 to produce the final output. Here, the concept of congestion is defined for systems with a two-stage structure. Then, to examine the congestion of each stage as well as the congestion of the whole system, a single linear programming model is proposed. The validity of the proposed model is investigated using several theorems and it is shown that the new definition is a generalization of the previous definitions of the congestion for the black-box systems. Finally, the proposed model is applied to a case study including 24 non-life insurance companies in Taiwan.

In some examinations, the changes in performance measures and the efficiency of entities should be addressed over a span of time despite the undesirable outputs and unsatisfactory convexity property. In this study, the performance of processes is analyzed over several periods of time by presenting a multi-period model based on the free disposal hull (FDH) approach, incorporating undesirable outputs. Using the product of availability, performance (efficiency) and quality, the overall equipment effectiveness (OEE) is calculated which shows the level of productivity. After estimating the performance and OEE, an inverse multi-period FDH model is introduced to measure the input changes for the changes made to outputs in different time periods. The proposed approaches are used in the automotive industry to analyze the performance and changes in input measures over multiple periods of time. The results show that the proposed technique is reasonable for assessing the multi-period efficiency, the period and overall OEE of the systems, and the input changes related to several periods while the convexity assumption is violated.

In resource allocation and target setting problems, a central decision makers’ managerial standpoint has a pivotal role, especially when we encounter undesirable outputs such as the greenhouse gas (GHG) emissions. In such circumstances, firms have to cooperate with each other, to achieve the central planner’s aims. Looking into literature reveals that the existing resource allocation models based on data envelopment analysis (DEA) have not aptly considered the influence of managerial efforts and technological innovations in this sense. This study proposes a centralized model incorporating managerial disposability. This model not only reflects the leadership performance of the central planner and the technological novelty perspective in the resource allocation and target setting problem, but also has a positive modification against an environmental adaptation change. In order to illustrate the applicability of our resource allocation and target setting model, a case study of 23 US and Canadian oil and gas companies has been conducted. Analysis of the results reveals the appropriacy and efficiency of our proposed model in dealing with the current perspectives concerning the issue of resource allocation and target setting.

Abstract In the nonparametric data envelopment analysis literature, scale elasticity is evaluated in two alternative ways: using either the technical efficiency model or the cost efficiency model. This evaluation becomes problematic in several situations, for example (a) when input proportions change in the long run, (b) when inputs are heterogeneous, and (c) when firms face ex-ante price uncertainty in making their production decisions. To address these situations, a scale elasticity evaluation was performed using a value-based cost efficiency model. However, this alternative value-based scale elasticity evaluation is sensitive to the uncertainty and variability underlying input and output data. Therefore, in this study, we introduce a stochastic cost-efficiency model based on chance-constrained programming to develop a value-based measure of the scale elasticity of firms facing data uncertainty. An illustrative empirical application to the Indian banking industry comprising 71 banks for eight years (1998–2005) was made to compare inferences about their efficiency and scale properties. The key findings are as follows: First, both the deterministic model and our proposed stochastic model yield distinctly different results concerning the efficiency and scale elasticity scores at various tolerance levels of chance constraints. However, both models yield the same results at a tolerance level of 0.5, implying that the deterministic model is a special case of the stochastic model in that it reveals the same efficiency and returns to scale characterizations of banks. Second, the stochastic model generates higher efficiency scores for inefficient banks than its deterministic counterpart. Third, public banks exhibit higher efficiency than private and foreign banks. Finally, public and old private banks mostly exhibit either decreasing or constant returns to scale, whereas foreign and new private banks experience either increasing or decreasing returns to scale. Although the application of our proposed stochastic model is illustrative, it can be potentially applied to all firms in the information and distribution-intensive industry with high fixed costs, which have ample potential for reaping scale and scope benefits.

Estimating the revenue efficiency of entities being evaluated is crucial as it provides valuable information about organizations, assuming that the output prices are known. This research introduces a new definition of optimal scale size (OSS) based on maximizing the average revenue efficiency (ARE). Additionally, the ARE is defined for both convex and non-convex sets, independent of returns to scale and the assumption that the vector of input-output prices of units is uniform. Moreover, to address the presence of uncertain data in real-world applications, the introduced ARE model is extended to evaluate systems with random inputs and outputs, along with approaches for its calculation. Finally, the proposed method is applied in an experimental example, calculating the ARE for a dataset of postal areas in Iran.IntroductionThe concept of optimal scale size has been extensively studied in the field of data envelopment analysis. Cesaroni and Giovannola's research on non-convex FDH technology reveals that the optimal scale size is a point in the production possibility set that minimizes average cost efficiency. Average cost efficiency, a new measure combining scale and allocation efficiencies, provides a more accurate performance assessment compared to cost and scale efficiencies. When evaluating units with known output prices instead of input prices, assessing revenue efficiency can offer more valuable insights. This paper extends the research on cost evaluation to revenue evaluation. It introduces the concepts of average revenue efficiency and optimal scale size based on revenue maximization. The optimal scale size based on revenue maximization is defined as the point in the production possibility set that maximizes the average radial income for the unit under investigation. Average revenue efficiency serves as an evaluation measure of unit revenue, surpassing revenue and scale efficiencies in accuracy. The paper examines methods for calculating average revenue efficiency in both convex and non-convex technologies. It demonstrates that the average revenue efficiency model in convex technology with variable returns to scale is equivalent to the revenue model with constant returns to scale. Furthermore, the relationship between optimal scale size points based on revenue maximization and the most productive scale size is determined. Next, the paper presents the average revenue efficiency model for stochastic sets with the presence of stochastic data. An experimental example is used to calculate the average revenue efficiency and obtain the optimal scale size for a set of postal areas in Iran.Materials and MethodsThe study builds upon Cesaroni and Giovannola's method for calculating average cost efficiency and optimal scale size to develop models for average revenue efficiency and optimal scale size based on revenue. It also utilizes chance-constrained probabilistic models with a deterministic objective function in DEA literature to present average revenue efficiency for stochastic sets. The model is transformed from stochastic to deterministic and then converted into a linear model using the error structure method.Discussion and ResultsThis paper introduces average revenue efficiency and optimal revenue scale size, demonstrating the equivalence between the average revenue efficiency models in convex technology with variable returns to scale and those with constant returns to scale. It also presents the average revenue efficiency model for stochastic sets, enabling the calculation of average revenue efficiency and optimal revenue scale size for units with random inputs and outputs.ConclusionIn many real-world scenarios, particularly when output prices are known, evaluating revenue efficiency holds greater significance than cost efficiency. This study develops the concepts of average cost efficiency and optimal scale size for revenue evaluation, expanding upon the existing literature on data envelopment analysis. The paper demonstrates how average revenue efficiency can be calculated as a valuable and accurate measure of efficiency in convex and non-convex technologies, without making assumptions about returns to scale. By assuming the randomness of input and output variables and employing chance-constrained models, a quadratic deterministic model is presented to calculate average revenue efficiency. It is then transformed into a linear model assuming uncorrelated variables, enabling the determination of average revenue efficiency and optimal scale size based on revenue maximization for random units. The proposed models are applied to a real-world sample, evaluating the average revenue efficiency of twelve postal units. The results highlight the models' ability to provide a more accurate evaluation of revenue efficiency and identify the best revenue scale size as the reference for inefficient units.