Returns To Scale Research Articles

Data envelopment analysis: From non-monotonic to monotonic scale elasticities

The concept of returns to scale (RTS) or local scale elasticities in data envelopment analysis (DEA)—stemming from variable returns to scale (VRS) technology—has been recently criticized because of its misbehavior in the case of decreasing returns to scale (DRS). Here, the instrument should imply a downsizing force for improving productivity. In classical VRS technologies, however, it can hide respective improvement potentials: the more, the larger a company is. The non-monotonic behavior of local scale elasticities can address this effect. This study shows this phenomenon does not apply when using multiplicative DEA models. Therefore, we propose a new global scaling index that works in the classical VRS technology. We prove the new index is weakly monotonic and illustrate our theoretical findings in a banking context.

Analysis of economic, production, and marketing aspects of potato farming in Changunarayan Municipality of Bhaktapur, Nepal

A comprehensive study was undertaken in the Changunarayan Municipality of Bhaktapur District to conduct a thorough cost-benefit analysis, analyze production functions, and evaluate the various marketing channels and associated challenges of potato cultivation. total of 100 farmers were surveyed, and gathered data were, analyzed using Excel and SPSS software. Despite an average cost per hectare of potato cultivation reaching NRs 370,662.58, which is comparatively higher than in other regions, the productivity was notably greater at 22.021 Mt/ha, surpassing the national average of 16.73 Mt/ha. With a benefit-cost ratio of 1.68, potato production was found economically viable in the region. The production function analysis, conducted using the Cobb-Douglas Model, revealed noteworthy insights among the six variables examined. Fertilizers, machinery, plant protection measures, and micronutrients were identified as underutilized inputs, as indicated by a RUE value surpassing unity. Conversely, labor cost and various other factors exhibited negative associations with average returns. The study also revealed that potato production in the area exhibited an increasing Return to Scale (RTS) value of 1.619, indicating that a 1% increase in input expenditure would lead to a 1.619% increase in returns. Farmers primarily choose a marketing channel with three intermediaries, which results in a larger market margin but a smaller share for the producers. Among the five key marketing challenges reported by farmers, the influence of intermediaries stands out as the most significant issue. In conclusion, this study highlights the pressing need for efficient marketing channels and fair pricing systems to support potato farmers in the region.

Performance evaluation considering academic misconduct of China’s higher education institutions

Performance evaluation is essential for managing and allocating resources in higher education institutions (HEIs), guiding their development. However, in the current evaluation system, the growing issue of academic misconduct (AM) as an undesirable output in research assessments is highly overlooked. To fill this gap, we propose the creative inclusion of AM as an undesirable output in the evaluation system. To validate the potentially questionable nature of the evaluation system without AM, we took China’s 32 HEIs within the period of 2016–2018 as an example, separately calculating efficiencies without and with AM through DEA approach. Then we use Wilcoxon test of paired samples and Pearson correlation coefficient confirming that there are increasingly significant differences between the two efficiencies and the two rankings in the sample period, respectively. Therefore, we believe that other performance evaluations for HEIs that do not incorporate AM are likely to be questionable, such as the influencing factors of research efficiency and returns to scale (RTS). Incorporating AM into the evaluation index system, we further explore the influencing factors of research efficiency in sample HEIs through Tobit regression model. Then we propose innovative models tailored specifically for analyzing RTS of HEIs in which undesirable outputs are under extended strong disposability and apply them to the empirical study of China’s HEIs. The results show that (1) AM is playing an increasingly prominent role in the research efficiency evaluation of HEIs; (2) local economic development, external exchanges and emphasis on AM in HEIs can improve research efficiency; however, government funding has a negative effect on it while the effect of human capital is not significant; (3) the investment scale in most China’s HEIs is optimize or excessive.

The Analysis of Data Preparation to Validate Model Values of Information Technology

Currently, there are some methods of preparing data for validating an IT value model correctly. One challenge in applying data mining to validate model values is to convert data into an appropriate form for this activity. Data mining algorithms can then be applied using the prepared data. The adequacy of data preparation often determines whether this data mining is successful or not. This study aims at creating a method for preparing the data during validation. The basic method used for data preparation is the Returns to Scale (RTS) method because it is easy to use and can be combined with further validation results. This method was applied by employing two models: two-factor and three-factor models. Both models are then compared to see the difference between them. The developed model is then tested on Branchless Banking (BB) and Downstream Petroleum (DP) industries. The results show that the method is applicable to prepare the data for validation. In addition, the results also demonstrate that both industries, DP and BB, have different result on data preparation, meaning that DP and BB have different ITs. This research contributes not only to a technique of preparing data for validating an IT value model by the RTS method but also can be a basis to work for data validation because it can give a result with the behaviour of the industry.

A Study on Port (Terminals) Operational Efficiency Analysis Using PCA-DEA Approach: Focusing on Container Terminals on Busan Port

Previous literature has recognised the importance of port efficiency to port performance, and two popular methodologies of DEA (Data Envelopment Analysis) and SFA (Stochastic Frontier Analysis) have been widely applied to efficiency evaluation. However, there seems a limitation of lacking objectivity when selecting the variables of inputs and outputs. Meanwhile, as for the largest port in Korea, although many related efficiency studies already exist for Busan Port, most of them are founded on a whole port level basis while seldom on the container terminal level. Therefore, to fill the gap, this paper introduces a PCA (Principal Component Analysis) - DEA approach to research the operational efficiency of nine container terminals of Busan Port with the latest data of 2021. The PCA-DEA port (terminal) evaluation approach is practical and unbiased, and it may have some special practical value for the port industry. Container throughput is set as the output, and 13 variables from three scopes of land, facilities and labour are identified as inputs, followed by PCA which eventually detects three indictors of “No. transfer crane (RMGC) in yard”, “Staff average salary” and “Container freight station (CFS) space” as the core inputs representing the above three dimensions. Then, the DEA (CCR) and DEA (BCC) models are used to gain the assessment results: “PNIT, “PNC” and “HJNC” are DEA efficiency, while others of “HPNT”, “BNCT”, “Jaseongdae”, “Shinseondae”, “Gamman” and “Shin Gamman” are non-DEA efficiency. In addition, to achieve the target efficiency for the non-effective terminals, this paper confirms improvements through Return to Scale (ROS) and input redundancy discussion. This study objectively evaluates port terminals and provides useful data when establishing strategies to strengthen port efficiency and competitiveness in the future.

Energy Intensity, Energy Efficiency and Economic Growth among OECD Nations from 2000 to 2019

This study examines the energy intensity (EI), energy efficiency (EE), and economic growth, measured by the type of returns to scale (RTS), of 37 nations in the Organization for Economic Co-operation and Development (OECD) from 2000 to 2019. We apply a non-parametric approach to estimate the three measures from their consumption of four primary energy sources, such as coal, gas, oil, and zero emission (e.g., renewable and nuclear power) as inputs and gross domestic product (GDP) as an output. In this study, we have the two types of efficiency measures over time: window-based and cross-sectional-based measures. Three findings are identified from our empirical study. First, the operationally efficient group, including France, Iceland, Japan, Switzerland, UK, and USA, presented a stable status of full efficiency in the window-based efficiency measure. Iceland and Switzerland were also in the higher efficiency group based on the cross-sectional measure. Their efficiencies were high and stable over the observed periods. Second, zero-carbon-emission (e.g., renewable and nuclear) energies outperformed other energy sources (coal, gas, and oil) in terms of a potentiality of EI/EE improvement. In other words, OECD nations can improve on their EI/EE measures by reducing fuel consumption of coal, gas, and oil while maintaining their high GDP levels. Finally, four industrial nations (France, Japan, UK, and USA) had a status of unity in their EI/EE measures for zero-carbon-emission energies with decreasing RTS. These nations would increase zero-carbon emission for energy consumption to increase GDP while keeping optimal EI/EE because such changes in consumption would not largely affect EI/EE due to their constant RTS status. Iceland showed increasing RTS. The nation may improve the EI level by increasing zero-carbon-emission energy consumption and economic size. The four nations can increase zero-emission energy consumption to achieve further economic growth without observing a large deterioration of EI/EE because it is very close to constant RTS. The examination of RTS provides policy directions for the improvement of EI and EE. Switzerland showed decreasing RTS and may deteriorate the EI/EE by increasing energy consumption and the size of each economy. The remaining countries, whose degree of EI/EE measures was less than unity, showed increasing or decreasing RTS. The examination of RTS provides important implications for energy policy to enhance the degree of EI/ EE.

α-returns to scale with quasi-fixed inputs: an application to Québec hospitals

ABSTRACT This paper focuses on the determination and estimation of the optimal size of hospitals. To determine the optimal size of a production unit, we use the measurement of returns to scale (RTS) at the decision-making unit level. When the RTS are constant, the unit is deemed of optimal size as the average total cost is at its minimum. As we deal with public service in a non-market environment, we have to take into account the fact that hospitals may not operate efficiently. To estimate the required production frontier we adapt the α-returns to scale method (a DEA type algorithm compatible with non-convexity of the production set) to include quasi-fixed factors. This methodology is applied to Québec hospitals at different points in time in order to capture the effect of the restructuring of the public health system over the last three decades. We conclude that by relying more on larger institution the scale efficiency of the public system has increased. However, in spite of the large reduction in the number of small hospitals and their replacement by very large structures, the movement may have gone too far, as most of the large institutions tend to exhibits decreasing returns to scale.

Measuring and Analyzing Operational Efficiency and Returns to Scale in a Time Horizon: Assessment of China’s Electricity Generation & Transmission at Provincial Levels

This study discusses the assessment of OE (operational efficiency) and RTS (returns to scale) over a time horizon. Many previous DEA (Data Envelopment Analysis) studies have discussed how to measure OE/RTS. However, their works did not consider the measurement over time. The important feature of the proposed approach is that our models are different from standard ones in terms of factor (inputs and outputs) unification. A problem with standard models is that they produce different efficiency measures for input and output orientations. Consequently, they yield different OE and RTS estimates depending upon which production factor is used for measurement. To handle the difficulty, we develop a new DEA formulation whose efficiency measure is determined after combining inputs and outputs, and then we discuss how to measure the types of RTS. The other methodological feature is that the proposed model incorporates a time horizon. As an empirical application, this study considers electricity generation and transmission across Chinese provinces from 2006 to 2019. The first key outcome is that the performance of China’s electricity generation and transmission system tends to improve with an annual growth rate of 0.45% across time. The second outcome is that, during the observed periods, China has more occurrences of decreasing rather than increasing RTS. As an implication, some provinces (e.g., Jiangxi and Hainan) need to increase their generation sizes to enhance their OE measures, while other provinces (e.g., Jiangsu and Zhejiang) should decrease their generation sizes. Finally, this study confirms significant technological heterogeneity across Chinese provinces and groups.

A two-stage improved Base Point Slacks-Based Measure of super-efficiency for negative data handling

For most of the Slacks-Based Measure (SBM)-type models in Data Envelopment Analysis, the variable returns-to-scale (VRS) condition is required to become feasible and translation invariant while applied on negative data. Unlike them, the “Base Point SBM (BP-SBM)” and “Super BP-SBM” models accept negative values under four returns-to-scale (RTS) conditions; however, (i) they are not translation invariant, (ii) they are sensitive to some perturbation terms which help to translate negative data to positive data by avoiding division-by-zero irrationality, and (iii) the Super BP-SBM model suffers from infeasibility issue. To address the above limitations, this study improves the existing BP-SBM and Super BP-SBM models. The Improved BP-SBM (IBP-SBM) and Improved Super BP-SBM (ISBP-SBM) models are translation invariant under four different RTS conditions, and do not involve any perturbation terms. Although the proposed ISBP-SBM model is feasible under the four RTS conditions, it fails to provide Pareto efficient projections. Therefore, we propose a two-stage super-efficiency approach, where the ISBP-SBM model is applied on the first stage to obtain the input savings and output surpluses, which are incorporated in the IBP-SBM model to be applied on the second stage. To validate the proposed models, their results are compared with some state-of-the-art models using a benchmark dataset. Experimental findings indicate that the proposed models have overcome the limitations of the existing models while applied on negative data. Further, three real-life case studies of supplier selection, financial performance evaluation, and airline performance evaluation are used to discuss the applicability of the proposed approach.

Efficiency modeling, an important factor in the development of sheep and goat production in a semiarid region / Modelagem da eficiência, fator importante para o desenvolvimento da ovinocaprinocultura, dos produtores do Semiáridoc

The paper aims to use the Data Envelopment Analysis (DEA) approach to evaluate the performance of sheep and goat production systems in the semi-arid region in Brazil. By applying a questionnaire with 127 variables, we obtained information from 254 producers in the municipality of Tauá - CE, with the input variables number of sheep and number of cattle, and the output variable of revenues from animal production. The DEA model with constant returns to scale (RCE) and variable returns to scale (RVE) and were used. The average technical efficiency measures of RCE and RVE models were 9.72% and 23.65% for all farmers. Only two and eight producers obtained efficiency equal to one (maximum) in RCE and RVE, respectively. Most producers operate in terms of increasing returns, and are inefficient. That is, the vast majority of producers are working below the optimum scale, which leads to the exclusion of producers.

Simulation designs for production frontiers

There are very few simulation experiments in the literature of multiple input-output production frontiers. Among these studies, simulations are usually designed for the case of one single output, according to their purpose of the studies. In this study, a simulation framework is proposed to examine whether a multiple input-output production frontier model is satisfactory, according to its objectives, in a controlled environment. The mathematical foundation is developed to generate multiple input-output production frontiers including different types of returns to scale (RTS) and inefficiency distributions. From the provided framework, the number of units, the number of inputs, the number of outputs, the number of efficient units, the number of inefficient units, the number of outliers, the RTSs, and the distribution of inefficiencies and outliers can be controlled. The performance of a theoretical production frontier model can then be examined on random datasets with multiple input-output variables.

How large of a grant size is appropriate? Evidence from the National Natural Science Foundation of China.

Under the current universal trend towards larger grant sizes in research funding systems, we focus on how large of a grant size is appropriate. We study the directional returns to scale (RTS) to assess whether current grant sizes are the most productive. We take the General Program of the National Natural Science Foundation of China (NSFC) as an example and select three samples of physics, geography and management for an empirical study. We find that the optimal input direction and the most productive grant size scale is different for the three disciplines; based on the current grant size, physics should not expand the grant size and team size input, geography should further increase the grant size to improve performance and management should further expand the team size rather than the grant size. In this paper, we demonstrate a new method to calculate the optimal direction, which is the lowest rate of congestion, according to the characteristics of the General Program. Based on these results, we also calculate the most productive scale size. This method has certain value for project management.

Analysis of farmers’ market power in the value chain of Arabica coffee in Lam Dong Province, Vietnam

The market that operates effectively will contribute to actors’ profit and welfare in the coffee value chain conversely if the market is affected by imperfect problems. Suppose the value chain lacks market news, short-term storage, information for product and processing, quality check, and market power. In that case, implied chain actors’ profit and welfare will not be reached. A new method can estimate market power by Lerner index through the stochastic cost frontier once the inputs data price is unavailable and the existence or non-existence of constant returns to scale. So this approach is applied to evaluate farmers’ market power in the value chain of Arabica coffee in Lam Dong Province, Vietnam. Primary data were collected through interviews with 200 farmers Da Lat, Lac Duong, Lam Ha, and Don Duong of Lam Dong Province. The estimated results of the market power and Lerner rate are 0.001. This index indicates that the local coffee market is perfect competition. So the coffee growers do not have market power. The RTS index of 0.56 (less than one) illustrates that farmers’ Return To Scale (RTS) is decreased. Therefore, coffee farmers should link together to establish cooperative that increases market power in price negotiation and coffee consumption.

Economic analysis of mushroom enterprise in Chitwan district, Nepal

The majority of the population (66%) in-country “Nepal” are engaged in agriculture. However, domestic production finds it difficult to meet the annual demand of the people. Hence, people are moving from subsistence agriculture to embrace mushroom farming. This study focuses on economic analysis and analysis of the present status of mushroom farming and enterprise in this country. The study was conducted in the land area of Kalika Municipality and Bharatpur Metropolitan City. 30 mushroom farmers with two huts and at least three years of experience were selected from the study area. The primary data were collected through face-to-face interviews with the farmers, focus group discussion (FGD) and key informant interviews (KII). The secondary data was collected through various published articles and documents. The data analysis was done using basic statistics and a regression function. The benefit-cost ratio is 2.54 and a high gross margin is NRs.490,876.65 per kattha per year. The return to scale (RTS) is 0.80. Five marketing channels are present among which wholesalers and local collectors contributed the highest percentage of the share. However, the dominance of the intermediaries, timely unavailability of inputs, price fluctuation, disease and pest infestation were the major constraints. Disease and pest control, formation of the producer organization, improvised cultivation practices, timely and affordable availability of quality can be the major solution measures. Whereas, suitable climatic conditions, high productivity and growing market demand are the strengths of mushroom production in this study area. Mushroom farming is found to be a profitable business concerning competitive and comparative markets.

Revenue efficiency evaluation of life insurance companies in India: identification of leaders and laggards

PurposeThe purpose of this article is to evaluate revenue efficiency performance of life insurance companies in India. The study also compares if private or public insurance sector is more “revenue efficient”. Furthermore, the study determines the nature of return to scale (RTS) and identifies the leaders and laggards amongst insurance companies operating in India.Design/methodology/approachRevenue efficiency is calculated by employing data envelopment analysis – a non-parametric approach, on a data set of 24 insurance companies over the period 2013–2014 to 2017–2018.FindingsThe empirical results suggest that life insurance companies in India could generate only 34.4% of revenue, which is very less than what these are expected to generate from the same inputs. Majority of life insurance companies operating in India are operating at decreasing return to scale (DRS). There is a reduction in leaders and the highest proportion of companies is falling in the category of laggards.Originality/valueAs per the best knowledge of researchers, no empirical work has been carried out with respect to measuring the revenue efficiency of Indian insurance companies. The current study appropriately fills the gap by not only calculating the revenue efficiency scores of insurance companies in India but also provides insights into the causes of revenue inefficiencies. It also gives implications for efficient and effective management of insurance companies.

Financial performance of the textile industry in India: the case of Haryana

PurposeThis study aims to evaluate the financial performance of the textile industry in Haryana located in the northern part of India.Design/methodology/approachInput-oriented Cooper, Charnes and Rhodes (CCR) and Banker, Charnes and Cooper (BCC) techniques of data envelopment analysis, as well as the return to scale (RTS) technique, were used to conduct the analysis.FindingsThe findings show that textile units in Haryana have hugely underperformed financially with a consolidated technical efficiency score of only 0.35. Both private and public limited textile companies with respective scores of 0.46 and 0.24 are technically efficient. Public limited textile companies are more efficient than private limited companies. Private limited textile companies need to increase their input scale because they are operating at an increasing return to scale while public limited textile companies have to lower their input scale because most companies are operating at a decreasing return to scale to enhance their efficiency.Originality/valueThe study can assist in decision-making to all key stakeholders (Shareholders, management, government, tax authorities, debtors and creditors, among others) by identifying efficient and inefficient companies. Appropriate policies can be framed based on that knowledge.

Analysis of Production Factors Influences of Pineapple Farming in Kualu Nenas Village Tambang Sub-District, Kampar Regency

Kualu Nenas Village is a center of pineapple production in Tambang District, Kampar Regency. Pineapple production and productivity for the last seven years in Tambang District were fluctuating and decreasing every year. Pineapple farming is the main source of income in the village and has not been managed properly by using its production factors. This study aims to describe the performance of pineapple cultivation by farmers and to analyze the effect of the number of production factors on pineapple productivity. This research was conducted in Kuala Nenas Village. The method of analysis using descriptive statistics and multiple linear regression analysis. The data used is time-series data in the form of cross-section data from pineapple production data in Kualu Nenas Village for a period of four months, starting from November 2019 to February 2020 from direct interviews with 35 farmers using the purposive sampling method. The results of the study conclude that the management of pineapple cultivation at of loosening the soil, weeding, controlling pests and diseases, fertilizing, giving ethrel, thinning tillers, and harvesting activities. The results of the analysis show that the variation in the total productivity of each farmer is 86,20percents due to differences in the use of production factors per hectare. Of the seven production factors that are thought to affect the productivity of the pineapple, only three production factors have a statistically significant effect, namely the use of pineapple staple population with a production elasticity of 0,398percents, the labor usage with a production elasticity of 0,325percents and the use of amount ethrel with a production elasticity of 0,103percents. While, four factors production which did not significantly influence the productivity of pineapples namely the amount of KCL fertilizer, urea fertilizer, ash fertilizer, and ZA fertilizer because the elasticity of production is less than 1 percent. The Return to Scale (RTS) condition of the business is in the Decreasing Return to Scale (DRS) condition with a value of 0,885.

Directional scale elasticity considering the management preference of decision-makers

Most data envelopment analysis (DEA) studies on scale elasticity (SE) and returns to scale (RTS) of efficient units arise from the traditional definitions of them in economics, which is based on measuring radial changes in outputs caused by the simultaneous change in all inputs. In actual multiple inputs/outputs activities, the goals of expanding inputs are not only to obtain increases in outputs, but also to expect the proportions of such increases consistent with the management preference of decision-makers. However, the management preference is usually not radial changes in outputs. With the latter goal into consideration, this paper proposes the directional SE and RTS in a general formula for multi-output activities, and offers a DEA-based model for the formula of directional SE at any point on the DEA frontier, which is straightforward and requires no simplifying assumptions. Finally, the empirical part employs the data of 16 basic research institutions in Chinese Academy of Sciences (CAS) to illustrate the superiority of the proposed theories and methods.

Non-radial two-stage network DEA model to estimate returns to scale

PurposeAs returns to scale (RTS) describes the long run connection of the changes of outputs relative to increases in the inputs, the purpose of this study is to answer the following questions: If the proportionate changes exist in the inputs, what is the rate of changes in outputs with respect to the inputs’ variations in the two-stage networks over the long term? How can the authors investigate quantitative RTS in the two-stage networks? In other words, the purpose of this research is to introduce a different approach to estimate the performance, RTS and scale economies (SE) in network structures.Design/methodology/approachThis paper proposes a novel non-radial approach based on data envelopment analysis to analyze the performance and to investigate RTS and SE in two-stage processes.FindingsThe findings show that the range adjusted measure (RAM)/RTS approach can identify reference sets for overall systems and each stage. In addition, the models presented in this paper can classify decision-making units and determine the increasing/decreasing trends of RTS.Originality/valueThe majority of previous RTS studies have been examined in black-box structures and have been discussed in a radial framework. Therefore, in this study, RTS and SE in the two-stage networks are dealt with using an extended RAM approach. Actually, the efficiency and RTS for each stage and the overall model are calculated using the proposed technique.

Evaluating performance of super-efficiency models in ranking efficient decision-making units based on Monte Carlo simulations

In response to the limitation of classical Data Envelopment Analysis (DEA) models, the super efficiency DEA models, including Andersen and Petersen (Manag Sci 39(10): 1261–1264, 1993)’s model (hereafter called AP model) and Li et al. (Eur J Oper Res 255(3): 884–892, 2016)’s cooperative-game-based model (hereafter called L–L model), have been proposed to rank efficient decision-making units (DMUs). Although both models have been widely applied in practice, there is a paucity of research examining the performance of the two models in ranking efficient DMUs. Consequently, it is unclear how close the rankings obtained by the two models are to the “true” ones. Among the very few studies, Banker et al. (Ann Oper Res 250(1): 21–35, 2017) pointed out that the ranking performance of the AP model is unsatisfactory; Li et al. (Eur J Oper Res 255(3): 884–892, 2016) and Hinojosa et al. (Exp Syst Appl 80(9): 273–283, 2017) demonstrated the L–L model’s capability of ranking efficient DMUs without addressing the ranking performance. In this study, we, thus, examine the ranking performance of the two super-efficiency models. In evaluating their performance, we carry out Monte Carlo simulations based on the well-known Cobb–Douglas production function and adopt Kendall rank correlation coefficient. Unlike Banker et al. (Ann Oper Res 250(1): 21–35, 2017), we use the rankings obtained based on the two models and the “true” ones as the basis of performance evaluation in our simulations. Moreover, we consider several types of returns to scale (RS) and study the impact of changes of some parameters on the ranking performance. In view of the importance, we also carry out additional simulations to examine the influence of technical inefficiency on the two models’ ranking performance. Based on the simulation results, we conclude: (1) Under different RS, the ranking performance of the two models remains the same when changing parameters, e.g., the distribution of input variables; (2) Under different RS, when technical inefficiency (in comparison with random noise) is more important, the two models have satisfactory performance by providing rankings that are close to, or the same as, the “true” ones; (3) The L–L model has better performance than the AP model and is more robust. This is especially true when technical inefficiency is less important; (4) Under different RS, when technical inefficiency is less important, both models have unsatisfactory ranking performance; and (5) The relative importance of technical inefficiency plays an prominent role in ranking efficient DMUs.