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A note on two-stage network DEA model: Frontier projection and duality
Abstract
In Chen, Cook, Kao, and Zhu (2013), it is demonstrated, as a network DEA pitfall, that while the multiplier and envelopment DEA models are dual models and equivalent under the standard DEA, such is not necessarily true for the two types of network DEA models in deriving divisional efficiency scores and frontier projections. As a reaction to this work, we demonstrate that the duality in the standard DEA naturally migrates to the two-stage network DEA. Formulas are developed to obtain frontier projections and divisional efficiency scores using a DEA model's and its dual solutions. The case of Taiwanese non-life insurance companies is revisited using the newly developed approach.
... Chen et al. [29] showed that multiplier and envelopment network models are two different methods for measuring the efficiency of two-stage network structures in Figure 1. Lim and Zhu [42] showed that the duality of standard DEA naturally transitions into model (1) for simple two-stage network process. They further point out that it is still unclear whether the same result can also be derived for general two-stage network processes. ...
... (10) (x io ,ẑ1 do ,ŷ 1 go ) in the Equations (10a)-(10c) forms the frontier of stage 1. Similarly, (x 2 ho ,ẑ 2 do ,ŷ ro ) in the equations (10d)-(10f) forms the frontier of stage 2. Lim and Zhu[42] showed that the frontier of the intermediate measure z do would be any choice such that z 1 do ≥ẑ do ≥ẑ 2 do inFigure 1. Similarly, there is a possible difference for z do in two stages: z 1 do ≥ẑ 2 do in Formula(10). ...
China’s economic development has achieved great success in recent years, but the problems of energy scarcity and environmental pollution have become increasingly serious. To enhance the reliability and efficiency between energy, the environment and the economy, sustainable development is an inevitable choice. In the context of measuring sustainability efficiency, a network data envelopment analysis model is proposed to formulate the two-stage process of energy use and operations management. A double frontier is derived to optimize the available energy for sustainable development. Due to nonlinearity, previous linear methods are not directly applicable to identify the double frontier and calculate stage efficiencies for inefficient decision-making units. To address this problem, this study develops the primal-dual relationship between multiplicative and envelopment network models based on the Lagrange duality principle of parametric linear programming. The newly developed approach is used to evaluate the sustainability efficiency of 30 administrative regions in China. The results show that insufficient sustainability efficiency is a systemic problem. Different regions should take different measures to conserve energy and reduce pollutant emissions for sustainable development. To increase sustainability efficiency, regions should support energy-saving and emission-reducing technologies in production processes and strengthen their capacity for technological innovation. Compared with energy use efficiency, operations management efficiency in China has a wider range of changes. During the operations management stage, there is not much difference between the capacity and quantity of each region. Based on benchmark regions at the efficiency frontier, there is an opportunity to improve operations management in the near future. Blockchain technology can effectively improve energy allocation efficiency.
... resources, i.e. the proportional relationship between input and output. The higher the degree of optimal resource allocation, the higher the efficiency [15]. The efficiency of urban green-energy development refers to the ability of urban green energy to obtain the maximum output or obtain an output with the minimum input cost. ...
... The model needs to meet the constraint advancement of the input-output model (see Equation (2)) and meet the constraint conditions based on economic growth factors and environmental constraints, i.e. Equation (15). ...
With the acceleration of urbanization, cities are the main targets for carbon neutrality and urban energy is the terminal of energy consumption and the integration point of various energy systems. Therefore, there is a need to promote the development of urban green energy and achieve low input and high output to achieve a low-carbon economy in cities. Previous studies have not considered the input–output efficiency of urban green-energy development. This study fills this gap. Based on the economic–energy–environmental framework, an input–output efficiency-evaluation index system for urban green-energy development was constructed. Based on improved data-envelopment analysis, a comparative evaluation of the input–output efficiency of green-energy development was carried out in 30 provinces in China in 2019. Considering the differences in regions, the development of urban green energy in different provinces was classified. From the perspective of a low-carbon economy, economic growth factors and environmental constraint factors were set. Together with the generalized Divisia index approach, the input–output efficiency optimization directions of urban green-energy development were obtained. The results showed that the input–output efficiencies of urban green-energy development in Jiangsu, Zhejiang, Fujian, Inner Mongolia, Ningxia and other provinces and cities were relatively high. Provinces with faster economic development and higher environmental carrying capacity have advantages after optimization and will become pilot areas for the development of urban green energy. This research provides a reference for the development of urban green energy in various provinces from the input and output perspective.
... . By this procedure, we transform the input multipliers with large value range to variables with value range [0, 1]. Liang et al. (2006) and Lim and Zhu (2016) discussed that the arithmetic mean-based two-stage DEA model is nonconvex and cannot be converted into linear programming. A HSP-based method developed by Liang et al. (2006) can be used to obtain the global optimal solution. ...
... Denote E d as the integrated efficiency of a two-stage production process. It can be evaluated via the following sum of two fractional efficiency models: As discussed by Liang et al. (2006) and Lim and Zhu (2016), the model cannot be converted into linear programming via Charnes-Cooper transformation. E 1 d is denoted as the efficiency of the first sub-process, i.e., ...
The majority of data envelopment analysis (DEA) models can be linearized via the classical Charnes–Cooper transformation. Nevertheless, this transformation does not apply to sum-of-fractional DEA efficiencies models, such as the secondary goal I (SG-I) cross efficiency model and the arithmetic mean two-stage network DEA model. To solve a sum-of-fractional DEA efficiencies model, we convert it into bilinear programming. Then, the obtained bilinear programming is relaxed to mixed-integer linear programming (MILP) by using a multiparametric disaggregation technique. We reveal the hidden mathematical structures of sum-of-fractional DEA efficiencies models, and propose corresponding discretization strategies to make the models more easily to be solved. Discretization of the multipliers of inputs or the DEA efficiencies in the objective function depends on the number of multipliers and decision-making units. The obtained MILP provides an upper bound for the solution and can be tightened as desired by adding binary variables. Finally, an algorithm based on MILP is developed to search for the global optimal solution. The effectiveness of the proposed method is verified by using it to solve the SG-I cross efficiency model and the arithmetic mean two-stage network DEA model. Results of the numerical applications show that the proposed approach can solve the SG-I cross efficiency model with 100 decision-making units, 3 inputs, and 3 outputs in 329.6 s. Moreover, the proposed approach obtains more accurate solutions in less time than the heuristic search procedure when solving the arithmetic mean two-stage network DEA model.
... Numerous methodologies in the two-stage network have focused on deriving the overall efficiency and decomposing overall efficiency to obtain divisional efficiency scores. For more details and finding a beneficial review in this field, the researcher refers the reader to Chen et al. (2009), Kao (2014a), Li et al. (2012), Cook and Zhu (2014), Lozano (2015), Kao (2014b), Sahoo et al. (2014), Chen et al. (2016), Lim and Zhu (2016), Kao and Hwang (2008), Liang et al. (2008), and Lim and Zhu (2016). ...
... Numerous methodologies in the two-stage network have focused on deriving the overall efficiency and decomposing overall efficiency to obtain divisional efficiency scores. For more details and finding a beneficial review in this field, the researcher refers the reader to Chen et al. (2009), Kao (2014a), Li et al. (2012), Cook and Zhu (2014), Lozano (2015), Kao (2014b), Sahoo et al. (2014), Chen et al. (2016), Lim and Zhu (2016), Kao and Hwang (2008), Liang et al. (2008), and Lim and Zhu (2016). ...
... Numerous methodologies in the two-stage network have focused on deriving the overall efficiency and decomposing overall efficiency to obtain divisional efficiency scores. For more details and finding a beneficial review in this field, the researcher refers the reader to Chen et al. (2009), Kao (2014a), Li et al. (2012), Cook and Zhu (2014), Lozano (2015), Kao (2014b), Sahoo et al. (2014), Chen et al. (2016), Lim and Zhu (2016), Kao and Hwang (2008), Liang et al. (2008), and Lim and Zhu (2016). ...
... Numerous methodologies in the two-stage network have focused on deriving the overall efficiency and decomposing overall efficiency to obtain divisional efficiency scores. For more details and finding a beneficial review in this field, the researcher refers the reader to Chen et al. (2009), Kao (2014a), Li et al. (2012), Cook and Zhu (2014), Lozano (2015), Kao (2014b), Sahoo et al. (2014), Chen et al. (2016), Lim and Zhu (2016), Kao and Hwang (2008), Liang et al. (2008), and Lim and Zhu (2016). ...
... Hospitals need appropriate access to resources not only in physical and non-physical respects ones but also in social ones. Data envelopment analysis (DEA) among all the existed methods to assess the efficiency has been noted and proven to be an effective approach in recent years [10][11]. DEA is a nonparametric method with no frontier about the efficiency boundary that evaluates the relative efficiency among a group of observed homogenous decision making units (DMUs) which can change over time in regard to comparable objectives as well as variables which can be grouped into two separate groups of inputs and outputs [10]. ...
... Data envelopment analysis (DEA) among all the existed methods to assess the efficiency has been noted and proven to be an effective approach in recent years [10][11]. DEA is a nonparametric method with no frontier about the efficiency boundary that evaluates the relative efficiency among a group of observed homogenous decision making units (DMUs) which can change over time in regard to comparable objectives as well as variables which can be grouped into two separate groups of inputs and outputs [10]. ...
... Network DEA models usually assume free disposability of excess output of intermediate products (e.g. Färe & Grosskopf (2000), Kao (2014), Lim & Zhu (2016); an exception is Tone & Tsutsui (2009, p. 246)). In case of Example 4.1 with a single intermediate product any excess output 0 > 0 can be reduced to zero ( 0 = 0) by decreasing certain activity levels of DMUs in stage A which does not lead to an increase of input of primary factors #1 and #2 whereas the production of stage B is not changed. ...
Production takes place in complex networks. An important question is how the efficiency of the whole network is related to that of the individual production units forming the network. A general network production theory with following characteristics is developed. Analysed networks may possess arbitrary structures with units whose technologies may be non-convex and even discrete. The theory generalises Koopmans’ linear activity analysis based on similar underlying modelling features and fundamental assumptions. The modelling approach is more suitable for analysing networks than common ones. Methods of efficiency measurement known from network data envelopment analysis are integrated into the theory. It is shown that calculating an overall efficiency score for a network as average of individual scores of its units is inappropriate. The relationship between the efficiency of a network activity and that of subsystems and units strongly depends on the extent of which the individual production units are free to choose their input and output quantities, i.e. whether the network is loose or tied. Especially in cases where flows of intermediate products are constrained instead of freely disposable, the explicit modelling of their overproduction helps to analyse their influence on efficiency scores.
... E.g.Färe & Grosskopf (2000),Kao (2014),Lim & Zhu (2016); in contrast,Tone & Tsutsui (2009, p. 246) do not assume free disposability for their intermediates. That the "inclusion of intermediate variables in the efficiency measure" should be a main line for future research is a consequence whichAlves and Meza (2023, p. 2746) draw as conclusion from their literature review. ...
Production usually takes place in complex networks. Network activity analysis is a recent approach to model and study networks of an arbitrary, possibly hierarchical structure that is composed by systematically linking its production units as subsystems. By embedding Koopmans' linear activity analysis into a broader framework the approach is generic as it requires rather weak technological properties, thus permitting non-convex and even discrete production possibilities. The present paper includes 'bads' as undesirable objects into network activity analysis, utilising a framework that integrates multi-criteria decision analysis and production theory. General and more specific theorems are derived and illustrated by nonlinear and linear technologies that relate important properties of networks (e.g. convexity) or of their activities (e.g. efficiency) to those of their units. By applying different types of efficiency measures, a systematic procedure for evaluating the performance of network activities is demonstrated for a two-stage production and abatement network and its subsystem with parallel units. Thereby, an inconsistency in assessing excess quantities of outputs, particularly of intermediates, is discussed which may arise with free disposability assumptions.
... Remark 4.3: Network DEA models usually assume free disposability of excess output of intermediate products (e.g. Färe and Grosskopf (2000), Kao (2014), Lim & Zhu (2016); an exception is Tone and Tsutsui (2009, p. 246)). In case of Example 4.1 with a single intermediate product any excess output 5 > 0 can be reduced to zero ( 5 = 0) by decreasing certain activity levels of DMUs in stage A which does not lead to an increase of input of primary factors #1 and #2 whereas the production of stage B is not changed. ...
Today, production usually takes place in complex networks. An important question is how the efficiency of the whole network is related to that of its units. Respective research on this topic has been strongly growing over the past decades, as a rule using methods of data envelopment analysis that are known as “network DEA”. However, there is a lack of theoretical foundation that allows clear statements to be made for arbitrary network structures and general, possibly non-convex or even discrete production technologies. This paper develops an activity analytic approach for modelling such general production networks and measuring their efficiency. Based on work of Koopmans and embedding it into a broader framework the approach is generic as it requires rather weak premises with regard to production technology and allows the network to be simply composed from its units as subsystems. It is shown that the relationship between the efficiency of a network activity and that of the subsystems and units depends strongly on the extent of which the individual production units are free to choose their input and output quantities, i.e. whether the network is loose or tied. Especially in cases where flows of intermediate products are constrained (instead of freely disposable), the explicit modelling of their overproduction helps to analyse their influence on efficiency scores. It is furthermore shown that calculating an overall efficiency score for a decision-making unit as average of individual scores of network units is inappropriate in any case.
(See DOI: 10.1007/s11573-025-01228-9 for a published revised version.)
... The study employed the slack variables of three input variables: the proportion of R&D personnel to enterprise employees, the proportion of employees holding postgraduate degrees or higher, and R&D expenditure investment. These were used as explanatory variables in constructing a SFA model (Devi et al., 2020;S. Lim & Zhu, 2016). The SFA utilized the total output value of the high-tech industry, the amount of technology transfer, and the level of digitization as explanatory variables. The effects of data and seven environmental variables on the three input slack variables were recalculated using Frontier 4.1 software (C. Guo & Zhu, 2017). Table 6 presents the r ...
This study investigates how the digital business environment affects firms' innovation input variables. It was discovered that digitization leads to ongoing corporate environment optimization, which improves the effectiveness of innovation. One of the institutional environment factors, digitalization, increases the redundancy of government subsidies on businesses' investments in innovation. It also helps to eliminate duplication in innovation investment through the financial environment and the protection of legal rights. With increasing marketization in the informal institutional framework, the degree of R&D investment redundancy lowers while R&D human resource investment redundancy grows. Digitization not only lowers the grade of innovation, but it also has a negative association with the duplicate nature of commercial R&D investments. The authors' research combines institutional environment theory and digital development to establish a new empirical foundation for corporate development in order to boost innovation efficiency.
... In other words, the number of cars purchased is a direct function of incentives and taxes, and an indirect function of CO 2 emissions. For this reason, the most appropriate empirical tool is the two-stage DEA model (see Chen et al., 2010;Cook et al., 2010;Chen et al., 2013;Lim and Zhu, 2016;Izadikhah et al., 2018). In our case, the choice over car models depends on the incentive to buy a green car and on the tax to be paid for cars featuring CO 2 emissions above a certain threshold. ...
... Huang and Eling [10] analysed the competence of the general insurance business in the four world's fastest-growing markets: Brazil, Russia, India, and China and concluded that the environment has a momentous impact on the non-life insurance market efficiency in BRIC countries. Lim and Zhu [14] discovered the dichotomy in the traditional DEA model, which shifts to a two-stage network DEA, and formulae are established to attain horizon estimates and divisional performance ratings utilising a DEA model and its duple solutions. Heng Chen. ...
In this paper a three-stage closed system fuzzy network Data Envelopment Analysis (DEA) is employed to assess the competency of reinsurers (both Indian and Foreign branch) functioning in India by disintegrating the complex service process into three sub-processes. The production, investment, and effectiveness frameworks are appraised in three stages. The output of the first stage is considered as the input for the second stage, and the output of the second stage is an input for the third stage. The fuzzy DEA findings reveal the individual efficiency score of production, investment , and the effectiveness sub-processes of each Decision-Making Unit (DMUs), which also ascertains the results of conventional DEA for all three stages. The data for this study has been taken from the Indian Insurance Regulator (IRDAI), spanning three years from FY 2017-2020.
... The projection of the intermediate measure in the centralized two-stage DEA model has been discussed by Chen et al. (2010) and Lim and Zhu (2016). They consider the projection of the intermediate measure to be any value within the interval ...
Performance evaluation of poverty alleviation (PA) is one of the important tools to promote the effective linkage between PA and rural revitalization in China. This paper exposes the internal structure of PA and specifies its input–output process as a two-stage series system consisting of public investment and PA processes. Based on the centralized and decentralized management mechanisms, we propose non-convex global two-stage data envelopment analysis (DEA) models and non-convex global Malmquist productivity indices (GMPIs) to measure the dynamic efficiency of each PA system and its changes. We select 22 provinces (regions) in China as decision making units (DMUs) to illustrate the validity of the proposed models. The results show that: (i) the global PA efficiencies of the overall system (subsystems) are increasing over time, and most of them have already been at a high efficiency in the recent period; (ii) there is a certain inconsistency in the performance development trends of the overall system and subsystems, particularly in terms of the non-convex GMPI, local efficiency change (LEC), and best practice change (BPC).
... Based on non-radial measures, especially slacks-based measure (SBM) (Tone, 2001), some other studies developed their approaches for basic two-stage network system (Chen et al., 2016), Although the SBM-based models in Kao (2018Kao ( , 2019 are proposed for closed series production systems and general network system, they are also suitable for two-stage network systems. Besides the efficiency measurement, a few studies investigated frontier projection and duality (Lim and Zhu, 2016;Chen et al., 2016). See reviews and references by Kao (2014Kao ( , 2017 for more comprehensive discussions on two-stage network DEA. ...
Two-stage network data envelopment analysis (DEA) is widely used to evaluate efficiency of different organizations with multiple operations processes or hierarchical structures. Although existing two-stage network DEA assumes two-stage systems resolve the inherent conflicts between two stages, the coordination effect between the two stages is usually ignored. Recently, the relation of two-stage network DEA to traditional “black box” DEA has been studied from the perspective of system coordination. A coordination efficiency was defined and measured by a DEA-based approach based on simple two-stage network systems. In this paper, we propose an extended DEA-based approach for measuring the coordination efficiency for general two-stage network systems. The paper shows that the coordination efficiency based on the multiplier DEA and envelopment DEA approaches is equivalent to each other under both constant returns to scale (CRS) and variable returns to scale (VRS) assumptions. The proposed approach is verified via two numerical examples finally.
... In fact, the process of science and technology innovation in industrial enterprises has obvious two-stage characteristics [73]. Thus, the two-stage network DEA model [74,75] is an effective method for considering difficulties in this regard. The two-stage network DEA model divides the "black box" into two stages for research. ...
This study develops a data-driven, comprehensive evaluation method to improve the science and technology innovative efficiency of industrial enterprises above designated size (hereinafter “industrial enterprises”). Based on an innovation value chain perspective, a two-stage evaluation index system is constructed. Thereafter, the Pearson correlation coefficient method was used to analyze correlations in the constructed index system. A two-stage network data envelopment analysis model with additional intermediate input was constructed to measure and evaluate industrial enterprises’ science and technology innovative efficiency from three aspects—research and development (R&D), commercialization, and comprehensive efficiencies—to reveal the temporal and spatial evolution. The feasibility and effectiveness of the method was verified using the statistical data of industrial enterprises in 16 cities in Anhui Province, China, from 2011 to 2020. The results show that the comprehensive efficiency of the scientific and technological innovation of industrial enterprises in these cities is at a medium level, and the efficiency development of the two stages is uncoordinated; the two-stage efficiency distribution tends to be “high R&D–high commercialization” and “low R&D–low commercialization”, and targeted countermeasures and suggestions are proffered. This study provides a reference for the sustainable development of industrial enterprises in relevant regions.
... Indeed, choosing between input-versus output-oriented targets, farthest versus closest targets (Aparicio et al., 2007;Kao, 2022) or targets located by parametrized directional distance functions (Chambers et al., 1996) are examples of the need to resort to managerial decisions in advance to locate the most appropriate target on the frontier. In NDEA, the problem is more intense as no matter which is the selected orientation, the targets of the intermediate measures can vary substantially (Lim and Zhu, 2016). In the case of the two possibly different targets obtained by the procedure Projection, the idea of "similarity" could be employed a posteriori to select the most appropriate one. ...
Network data envelopment analysis (NDEA) is an extension of standard data envelopment
analysis (DEA) that models the efficiency assessment of DMUs by considering their internal structure. While in standard DEA the DMU is regarded as a single process, in NDEA the DMU is viewed as a network of interconnected sub-processes (stages, divisions), where the flow of the intermediate products (measures) is essential in the efficiency assessment. In the prevalent conventional methodological approach to NDEA, the sub-processes are assumed as distinct entities with distinct inputs and outputs. Thus, each sub-process has its own production possibility set (PPS), which can be derived axiomatically from a set of assumptions using the minimum extrapolation principle. The PPS of the overall system is defined as the composition of the individual PPSs. The conventional approach comprises all the methods, in which the system and the divisional efficiencies are computed jointly in a single mathematical program. A fundamental property connecting the system with the divisional efficiencies is that a system
is overall efficient if and only if its divisions are all efficient. In NDEA, regardless of the
method used, there are cases where none of the observed DMUs is rendered overall efficient, as corroborated by real-world case studies. This is the main issue we discuss in this paper and the motivation to propose an alternative, non-conventional, approach to address it in the frame of two-stage processes. We consider the two-stage process as a system that can be viewed in two perspectives depending on the role of the intermediate measures: the system as producer and as consumer of the intermediates. As our approach is based on standard DEA, it satisfies the basic desirable properties. The fundamental NDEA property, that the overall system is efficient if and only if both perspectives are efficient, is met. The efficient frontier of the system is explicitly defined by the overall efficient observed DMUs and the inefficient DMUs are projected on the efficient frontier of the system. A convergent procedure is presented for this
purpose. The proposed models are equivalently expressed in both the multiplier and the
envelopment forms due to strict primal-dual correspondence and are able to operate under both constant and variable returns-to-scale assumptions. We use the case of twenty-two automotive manufactures for the fiscal year 2019 as an example to illustrate our approach. Comparison with other NDEA methods is also provided.
... The method was developed by [41] with the assumption of constant returns to scale (CRS), formerly extended to the variable return to scale (VRS) by [42]. In addition to the traditional forms of DEA, the NDEA is more attractive for researchers aiming to evaluate the efficiency of complex production models (with more than a singular stage of production), such as higher education [43], banking [44,45], insurance [46,47], healthcare systems [48,49], and so forth. ...
Considering the socio-economic importance of Incheon International Airport, this study explored the changes in its aeronautical and non-aeronautical efficiency between 2001 and 2021. The study was conducted to measure and observe the changes in efficiency during the pre- and post-pandemic era of COVID-19. We employed a two-stage analytical approach to obtain the results using a set of desirable and undesirable variables. For the first stage, we employed a novel network data envelopment analysis–window analysis model to find the efficiency measures; for the second stage, we applied the Tobit regression analysis to observe the impact of some control variables on efficiency levels. The empirical results from the efficiency analysis stage revealed that, although the pandemic negatively affected the efficiency of this airport, the gain from appropriate strategies mitigated the excessive efficiency decline. Moreover, aeronautical activities showed better efficiency than non-aeronautical activities during the study period. In addition, further investigation of the second-stage analysis implied that an outbreak of pandemic diseases such as COVID-19 would dramatically impact international hubs such as Incheon International Airport; however, focusing on the import and export activities, in addition to increasing the connectivity with other airports, would improve the efficiency.
... Then, in the input-oriented case, the frontier projection point can be calculated using the optimal solution as follows: (5) where improvement targets of intermediates can be any choice such that (Lim and Zhu, 2019). As Lim and Zhu (2016) stated, would be an easy choice for . In the application of this study, however, we prefer to proceed with the formula . ...
Banks constitute approximately 90% of the Turkish financial system, so an efficiently operating banking sector is essential for financial consolidation. To ensure the efficient functioning of the banking production process, the capital adequacy ratio (CAR), which is a basic indicator in controlling financial risk, should be managed properly. Additionally, the production process of banks fits into a typical two-stage system, thus opening the black-box on bank efficiency is necessary for accurate efficiency measurement. By focusing on the link between efficiency, risk, and return, this study aims to present a two-stage efficiency evaluation of the commercial banks in Turkey for the year 2018. In addition to the efficiency scores, frontier projections are determined, and an examination is made on the CAR targets. The empirical findings indicate that the inefficiency in the Turkish banking sector mainly stems from operational performance and the average efficiency score of the state-owned banks is the highest. According to the target values, a pattern is detected between the efficiency scores and CAR. We also conclude that the minimum capital adequacy of 10.5% set by Basel III is not high to guide the commercial banks in Turkey to the efficient frontier.
... For example, [70] argued that an identical optimal solution does not exist between mul-tiplier and envelopment forms of network DEA models. Lim and Zhu [71] counter-argued their claim and pointed out that the duality relation from the standard DEA naturally migrates to the twostage network DEA with a unique solution. In the robust optimization, Park's revelation had been established in Beck & Ben Tal [4] , who showed that the dual of the robust counterpart is not equivalent to the robust counterpart of the dual model. ...
Robust Data Envelopment Analysis (RDEA) is a DEA-based conservative approach used for modeling uncertainties in the input and output data of Decision-Making Units (DMUs) to guarantee stable and reliable performance evaluation. The RDEA models proposed in the literature apply robust optimization techniques to the linear and conventional DEA models which lead to the difficulty of obtaining a robust efficient DMU. To overcome this difficulty, this paper tackles uncertainty in DMUs from the original fractional DEA model. We propose a robust fractional DEA (RFDEA) model in both input and output orientation which enables us to overcome the deficiency of existing RDEA models. The linearized models of the fractional DEA are further used to establish duality relations from a pessimistic and optimistic view of the data. We show that the primal worst of the multiplier model is equivalent to the dual best of the envelopment model. Furthermore, we show that the robust efficiency in the input- and output-oriented DEA models are still equivalent in the new approach which is not the case in conventional RDEA models. We finally present a study of the largest airports in Europe to illustrate the efficacy of the proposed models. The proposed RDEA is found to provide an effective management evaluation strategy under uncertain environments.
... Li et al. [66] extended the relational two-stage DEA model to the VRS version and assessed the performance of participants in the Olympic game. Lim and Zhu [67] showed that the duality in the conventional DEA model is also held in the two-stage network DEA models. Izadikhah and Farzipoor Saen [68] proposed a twostage range directional measure with negative data to measure the sustainability efficiency of SCs. ...
Owing to today’s highly competitive market environments, substantial attention has been focused on sustainably resilient supply chains (SCs) over the last few years. Nevertheless, very few studies have focused on the efficiency evaluation analysis of the sustainability and resilience of SCs as an inevitable essential in any profitable business. This study aims to address this issue by proposing a novel fuzzy chance-constrained two-stage data envelopment analysis (DEA) model as an advanced and rigorous approach in the performance evaluation of sustainably resilient SCs. To the best of our knowledge, the current study is pioneering as it introduces a new fuzzy chance-constrained two-stage method that can be used to undertake the deterministic non-fuzzy programming of the proposed model. The proposed approach is validated and applied to evaluate a real case study including 21 major public transport providers in three megacities. The results demonstrate the advantages of the proposed approach in comparison to the existing approaches in the literature.
... Many studies have been proposed to measure performance using DEA models [46]. However, few of them address the problem of airport performance [29]; only one of them evaluates future airport performance for airport sustainability planning in short-term, and neither of them estimates the efficiency or performance of future airports infrastructures projects in mid and long term. ...
The aim of this paper is to design an approach to evaluate the expected efficiency and
performance of future airport infrastructure. First, an airport sampling method to select similar airports is developed based on socioeconomic and operational airport variables that are summarized in a proxy variable; second, the ARIMA-GARCH-Bootstrap method is applied to forecast the selected outputs (PAX and ATMS) whilst the selected inputs (Cities, Gates, Runaways, Airport Size, Pax carriers, and Num. of employees) remain constant; and third, the VRS-OO and the CRS-OO DEA models are implemented to evaluate the efficiency and performance of the airports in the current and future years. The proposed approach is used to evaluate the future airport infrastructure of the new
Mexico City Airport against 19 representative worldwide airport hubs. The proposed approach is applied to analyze the Mexico City Airport multi-airport system infrastructure as a case study. The results show that this multi-airport system requires more airside infrastructure that must be added by the new Mexico City Airport, airlines should operate aircrafts with more capacity to serve more PAX per ATM, and airlines must open new connections at the new Mexico City Airport to increase the expected efficiency and performance of this multi-airport system.
... Based on the primal-dual relationship of the two models, the envelopment model is reformulated to be able to obtain the projection point and measure the division efficiencies at the same time. Lim and Zhu [12,13] showed that theoretically the divisional efficiency scores and frontier projections can be calculated for the multiplicative and additive models by using the primal-dual correspondence of linear programming under basic two-stage network structure. Guo et al. [6] developed envelopment network model in parametric linear form to determine frontier projection and to find divisional efficiency. ...
In conventional DEA models it is assumed that each measure has to be identified whether it is an input or output. Nevertheless, in some cases flexible measures can exist which can play the role of either an input or output. In all of the previous studies did not point to classify flexible measures in two-stage network DEA. In this paper we propose new radial models are based on envelopment form of CCR such as non-oriented models under variable returns to scale, input- and output-oriented models under constant returns to scale in basic two-stage network structure. Our models can select the status of each flexible measure as an input or output and determine the relative efficiency and an efficient projection unit. The input- or output-oriented model under CRS in basic two-stage network structure may produce different efficiency scores and it can be expected that a flexible measure is selected as an input variable in one model but an output variable in another. On the other hand, the status of one flexible measure in two CRS and VRS environments may have different results. The difference between the results of the two models clearly shows the need for a model that considers VRS in the presence of flexible measure. Therefore, the main focus of this study was to propose a new non-oriented model and aggregate model that can determine a unique status for flexible measures under VRS in basic two-stage network structure. Numerical examples are used to illustrate the procedures.
... In the study of multi-input and multi-output problems, models in which statistical methods are employed to automatically assign weights can effectively reduce the impacts of the subjective index weighting method. As an evaluation method that comprehensively considers the relative efficiencies of inputs and outputs among decision-making units (DMUs), DEA is consistent with the core idea of eco-efficiency, and is widely used to assess industrial eco-efficiency (Lim and Zhu 2016). ...
Industry is the largest sector for energy consumption and pollution emissions in China. Thus, improving industrial eco-efficiency is necessary for China to achieve sustainable development. Based on panel data from 31 industrial sectors from 2001 to 2015, a three-stage data envelopment analysis model was used to empirically explore industrial eco-efficiency and its influencing factors from the perspective of industrial heterogeneity. The results show that the overall level of industrial eco-efficiency in China is not high, first declining and then rising during the study period. Low eco-efficiency was mainly due to low scale efficiency. After removing the influences of external environmental factors and noise, industry profit rates, ownership structures, and foreign direct investments were all significantly and positively correlated with eco-efficiency. Environmental regulations were significantly and negatively correlated, while the intensity of research and development exhibited no linear relationship. Industrial heterogeneity significantly affects eco-efficiency. Capital-intensive industries had the highest eco-efficiencies, followed by resource-intensive industries and labor-intensive industries, respectively.
Graphic abstract
Comparison of technical efficiency (TE) before and after adjustment on a panel of 31 industries in China from 2001 to 2015.
... In this case, Fare and Grosskopf (2000) proposed the Network DEA method, where a network consists of subtechnologies. The general structure of the network model allows us to apply it to a variety of situations, including intermediate products, allocation of budgets or fixed factors and certain dynamic systems (Kao and Hwang, 2008;Lim and Zhu, 2016;Tone and Tsutsui, 2017;Fukuyama and Matousek, 2017;Li et al., 2018a, b). Additionally, Tone (2001) proposed a slacks-based measure (SBM) of efficiency in DEA, which dealt directly with the input excesses and the output shortfalls of the DMU concerned. ...
Purpose
From the perspective of cause and effect, the operational processes of property insurance companies can be considered as historical events. The purpose of this study is to measure the operating efficiency of China's property insurance industry, explore the determinants that affect technical efficiency and outline the path to achieving high-quality development.
Design/methodology/approach
We chose 44 Chinese property insurance companies as research objects. The data were obtained from the Chinese Insurance Yearbook and China Statistical Yearbook 2015–2017. First, the data envelopment analysis (DEA) method was used to calculate the technical efficiency of property insurance companies. Then, Tobit regression and quantile regression were adopted to explore the influencing factors of technical efficiency. Finally, the fuzzy-set qualitative comparative analysis (fsQCA) method was employed to summarize the path to improving the operating efficiency of property insurance companies.
Findings
The empirical results in the first stage suggested that the operation efficiency of China's property insurance industry was technically inefficient, and the scale efficiency was relatively better than the pure technical efficiency. In the second stage, we observed that the drivers for firm size, reinsurance rate, claim ratio and equity restriction were important determinants of an insurance firm's efficiency.
Research limitations/implications
We also put forward four applicable, targeted and proven ways to improve the technical efficiency of property insurance companies. These configurations are verified by cases of existing property insurance companies, which can provide practical references for the insurance industry.
Originality/value
Our research enriches the insurance literature and efficiency methods, particularly regarding the specific paths of improving the technical efficiency. The relationship between elements and results is analyzed from a systematic perspective, and the research results are not only more consistent with what logic might imply but also more instructive for the improvement of reality.
... DEA is widely used in energy and eco-efficiency assessment [21,22]. Among the DEA models, BCC model is a basic method for efficiency assessment [23], but it cannot distinguish DMUs with the value of 1, while the super-efficiency DEA can effectively solve this problem [24]. ...
The industrial sector is the largest energy consumption sector and major source of environmental pollution in China. Improving the industrial eco-efficiency is significant for China to realize high-quality development. Based on the panel data of industrial sector in China’s 30 provinces from 2006 to 2015, this paper first adopts a dynamic network data envelopment analysis to evaluate the eco-efficiency of industrial sector, and then uses panel Tobit regression model to investigate the impacts of industrial policies on the eco-efficiency of industrial sector. The results reveal that, (1) the overall eco-efficiency of China’s provincial industrial sector is 0.7581 during 2006–2015, and the average efficiency of the industrial production process is higher than that of the waste treatment process. (2) The eco-efficiency of the industrial sector differs greatly by region, the eastern region is the highest, followed by the western and central regions, and the northeast is the lowest. (3) Industrial agglomeration impedes the improvement of industrial eco-efficiency, while industrial structure rationalization can promote industrial eco-efficiency. (4) The influence of industrial agglomeration and industrial structure rationalization on the efficiency of industrial waste treatment process is larger than that of the industrial production process. The findings of this paper reveal the importance of improving the efficiency of industrial sub-processes, and can also offer lessons for other industrialized nations to improve the industrial eco-efficiency.
... However, these studies mainly focus on applying DEA models. Only a few studies have mentioned the reasons why they employ a two-stage DEA method to evaluate operational efficiency or focus on the model's development [14][15][16][17][18][19][20][21]. Guo et al. [22] investigated the factors affecting the overall efficiency of the additive network DEA model. ...
Since Charnes, Cooper, and Rhodes introduced data envelopment analysis (DEA) in 1978, later called the DEA-CCR model, many studies applied this technique to different fields. Based on the original CCR model, many modified DEA models were developed by researchers. Since 1999, Seiford and Zhu presented a two-stage DEA model. Later, these models were widely used in many studies. However, the relationship between the efficiency scores that are obtained from the original CCR model and the two-stage DEA model remains unknown. To fill this gap, this study proposed a theoretical relationship between the efficiency scores that are calculated from the two-stage DEA model and those that are obtained from the original CCR model. How the sets of nonsymmetrical weights affected the efficiency scores were also investigated. Theorems regarding the relationship were developed, and then the model was utilized to evaluate the two-stage efficiency scores of the insurance companies (non-life) and bank branches. The results show that using a two-stage DEA model can get more information about operational efficiency than the traditional CCR model does. The findings from this study about the two-stage DEA technique can provide significant reasons for using this model to evaluate performance efficiency.
This chapter reviews the research on efficiency and productivity in the insurance industry utilizing frontier methodologies that have been published from 2012 to 2022. Forty-six empirical papers are surveyed. In these analyses, econometric and nonparametric approaches have been used to estimate frontiers, with data envelopment analysis (DEA) being the most frequently utilized method. The reviewed papers have been grouped into 12 different application areas where the main analyzed issues and/or tested hypotheses, as well as the principal findings, have been discussed.
The insurance industry plays a crucial role of the national financial system, and the operating efficiency of insurance companies has always been a significant subject of academic research. This study proposes a multiperiod DEA model to dynamically evaluate the operating efficiency of insurance companies, which not only overcomes the defect of the traditional DEA method ignoring the internal structure of decision-making units, but also extends the limitation of the leader–follower model in evaluating single-period efficiency. By analyzing the efficiency of seven listed insurance companies in China from 2009 to 2018, the following conclusions are drawn. The multiperiod DEA model demonstrates advantages over the leader–follower model. The profitability of insurance companies during the first five years is higher compared with the last five years. There is a significant correlation between premium financing efficiency and overall efficiency. Throughout both periods, the loss ratio, loss reserve ratio, and consumer price index (CPI) are always positively correlated with the efficiency of the insurer, while the gearing ratio is negatively related to the efficiency of the insurer. The correlation among gross domestic product (GDP), total insurance value, tradable financial assets, and corporate efficiency varies over time.
This paper addresses the efficiency measurement of firms composed by multiple components, and assessed at different decision levels. In particular it develops models for three levels of decision/production: the subunit (production division/process), the DMU (firm) and the industry (system). For each level, inefficiency is measured using a directional distance function and the developed measures are contrasted with existing radial models. The paper also investigates how the efficiency scores computed at different levels are related to each other by proposing a decomposition into exhaustive and mutually exclusive components. The proposed method is illustrated using data on Portuguese hospitals. Since most of the topics addressed in this paper are related to more general network structures, avenues for future research are proposed and discussed.
The simplest network structure is that of two divisions, in that the operations of the system are divided into two parts, performed by the two divisions. This type of system was first noticed by Charnes et al. (1986) in studying the performance of US Army recruitment. They found that army recruitment actually had two stages: creating awareness through advertising and signing contracts. To assess the impact of an input on the performance of the system, it is necessary to know the division that this input is associated with, so that the true effect of this input can be identified.
Network data envelopment analysis (NDEA) is an extension of standard DEA that models the efficiency assessment of decision‐making units (DMUs) by considering their internal structure. While in standard DEA, the DMU is regarded as a single process, in NDEA, the DMU is viewed as a network of interconnected sub‐processes (stages, divisions), where the flow of the intermediate products (measures) is essential in the efficiency assessment. In the prevalent conventional methodological approach to NDEA, the sub‐processes are assumed as distinct entities with distinct inputs and outputs. Thus, each sub‐process has its own production possibility set (PPS), which can be derived axiomatically from a set of assumptions using the minimum extrapolation principle. The PPS of the overall system is defined as the composition of the individual PPSs. The conventional approach comprises all the methods, in which the system and the divisional efficiencies are computed jointly in a single mathematical program. A fundamental property connecting the system with the divisional efficiencies is that a system is overall efficient if and only if its divisions are all efficient. In NDEA, regardless of the method used, there are cases where none of the observed DMUs is rendered overall efficient as corroborated by real‐world case studies. This is the main issue we discuss in this paper and the motivation to propose an alternative, non‐conventional, approach to address it in the frame of two‐stage processes. We consider the two‐stage process as a system that can be viewed from two perspectives depending on the role of the intermediate measures: the system as producer and as consumer of the intermediates. As our approach is based on standard DEA, it satisfies the basic desirable properties. The fundamental NDEA property, that the overall system is efficient if and only if both perspectives are efficient, is met. The efficient frontier of the system is explicitly defined by the overall efficient observed DMUs, and the inefficient DMUs are projected on the efficient frontier of the system. A convergent procedure is presented for this purpose. The proposed models are equivalently expressed in both the multiplier and the envelopment forms due to strict primal‐dual correspondence and can operate under both constant and variable returns‐to‐scale assumptions. We use the case of 22 automotive manufacturers for the fiscal year 2019 as an example to illustrate our approach. Comparison with other NDEA methods is also provided.
The last decade has witnessed an exponential proliferation of studies on Network Data Envelopment Analysis (NDEA) as a tool to measure efficiency and productivity for production systems. Those systems are composed of various layers of decision making (hierarchically organized) and potentially interconnected production processes. The decision makers face the problem of allocating resources to the various production processes in an efficient manner. This chapter provides a historical perspective to these developments by linking them to earlier works dating back to Kantorovich (Mathematical methods of organizing and planning production. Leningrad University, 1939) and Koopmans (Activity analysis of production and allocation, 1951). Both the allocation problem and the measures of efficiency used by these early authors are astonishingly relevant and similar to those in the recent NDEA literature. The modern researcher in NDEA should take stock of this early forgotten contributions.
Bankacılık operasyon sürecinin birbiriyle ilişkili alt süreçlerden oluşan network sistemlere uyduğu bilinmektedir. Tek aşamalı sistemlere kıyasla daha karmaşık bir yapıya sahip olan bu tür sistemlerin etkinlik analizinde standart VZA yaklaşımını kullanarak alt süreçleri göz ardı etmek yanıltıcı bulgular üretebilmektedir. Öte yandan, üretim ve aracılık yaklaşımları arasındaki çatışmadan dolayı mevduatın banka operasyon sürecindeki rolüne ilişkin literatürde bir fikir birliği bulunmamaktadır. Mevduatın ikili rolünün sorgulanması ile yola çıkılan bu çalışmada, potansiyel derinliğini henüz yansıtmamış ve mevduata yüksek oranda bağımlı olan Türk bankacılık sektörünün etkinliği üzerine kesitsel bir inceleme gerçekleştirmek amaçlanmaktadır. Bu doğrultuda, banka mevduatı bir ara değişken olarak tanımlanmakta ve birinci aşama üretim etkinliğini, ikinci aşama ise aracılık etkinliğini temsil etmek üzere iki-aşamalı bir network Veri Zarflama Analizi (VZA) modelinden yararlanılmaktadır. Türkiye'de faaliyet gösteren 23 mevduat bankasının 2020 yılı verilerinin kullanıldığı analizde, bulgular sektördeki ortalama %26'lık etkinsizliğin çoğunlukla üretim etkinsizliğinden kaynaklandığını ve standart VZA yaklaşımının etkinlik skorlarını olduğundan fazla tahminlediğini ortaya koymaktadır. Buna ek olarak, birinci ve ikinci aşama etkinlik skorlarına göre bankalar dört gruba ayrılmakta, her bir grup ve banka türü için hedef değerler hesaplanmakta ve gruplar arasındaki benzerlikler/farklılıklar hakkında değerlendirmeler yapılmaktadır.
Abstract It is known that the banking operation process fits into the network systems consisting of interrelated sub-processes. In the efficiency analysis of such systems, which have a more complex structure compared to single-stage systems, ignoring sub-processes by using the standard DEA approach can produce misleading findings. On the other hand, there is no consensus on the role of deposits in the bank operation process due to the conflict between the production and intermediation approaches. In this study, which is stimulated by questioning the dual role of deposits, it is aimed to perform a cross-sectional analysis on the efficiency of the Turkish banking sector, which has not yet reflected its potential depth and is highly dependent on deposits. Accordingly, bank deposit is defined as an intermediate variable and a two-stage network Data Envelopment Analysis (DEA) model in which the first stage represents operational efficiency while the second stage represents intermediation efficiency is used. In the analysis based on the 2020 data of 23 deposit banks operating in Turkey, the findings reveal that the average 26% inefficiency in the industry is mostly related to the operational inefficiency, and the standard DEA approach overestimates the efficiency scores. In addition, banks are divided into four groups according to the first and second stage efficiency scores, target values are calculated for each group and bank type, and evaluations are made on the similarities/differences between the groups.
Data envelopment analysis (DEA) is a technique to measure the performance decision making units (DMUs). Conventional DEA treats each DMUs as black-boxes, and the internal structure of DMUs is ignored. The two-stage DEA models as a special case of network DEA models explore the internal structures of DMUs. Most often, the data may expressed as ratio output/input and/or an output can not be produced by a certain input data. In these cases, traditional two-stage DEA models can no longer be used. It gave us the motivation to apply DEA-Ratio (DEA-R) for evaluate two-stage DMUs instead of traditional DEA to deal with these situations. For this aim, this paper develops two novel DEAR models namely, Range Directional DEAR (RDD-R) and (weighted) Tchebycheff norm DEAR (TND-R) models. The validity and reliability of our proposed approaches are shown by some examples. The Taiwanese non-life insurance companies is revisited using these proposed approaches and the results from the proposed methods are compared with those from some other methods.
Background
Grassroots CDC laboratory efficiency can reflect and influence a country's capacity for disease control and prevention, improving the efficiency of the grassroots CDC laboratory is very important to national CDC system, this article is based on DEA method, on the basis of clearly defining the concept of efficiency, more methods to build grassroots CDC laboratory efficiency evaluation index, taking China as an example, the grassroots CDC laboratory efficiency study, discusses the current problems in the development of grassroots CDC laboratory operation efficiency.
Methods
Using data from China National Health Development Research Center, combined with some public data from National Bureau of Statistics and China Health Statistics Yearbook data, use data envelopment analysis (DEA) to analyze the operation of China's grassroots CDC disease control laboratories from 2017 to 2019 Efficiency, input-output improvement value and total factor productivity changes.
Results
From 2017 to 2019, the operational efficiency of grass-roots CDC laboratories in China showed a downward trend, and there was a serious problem of insufficient output, especially the grass-roots CDC laboratories in western China, which had the lowest operational efficiency. By analyzing the change of total factor productivity, it is found that the total factor productivity of grass-roots CDC laboratories in China generally shows a downward trend, and the technical level and management efficiency have declined to a certain extent, and the working ability of CDC laboratories has not been effectively improved.
Conclusions
The operational efficiency of grass-roots CDC laboratories is declining, and the technical level and management efficiency are declining. The unreasonable utilization of laboratory resources and unbalanced development are still outstanding, and the operational efficiency of laboratories needs to be further developed. Policymakers should pay attention to the imbalance of resource utilization and development of grass-roots CDC laboratories, give certain policy support to inferior CDC laboratories, and urge grass-roots CDC laboratories to strengthen management, improve technical level and management efficiency, increase the actual output of laboratories, and further enhance the working ability of grass-roots CDC laboratories.
Performance or efficiency evaluation is of great importance for effective supply chain management, especially in the context of sustainable development and platform economy. In existing literature, two-stage network data envelopment analysis (DEA) has been widely used for supply chain efficiency evaluation. Although existing supply chain DEA usually assumes two-stage systems resolve the inherent conflicts between two stages, e.g. the supplier and manufacturer in a two-echelon supply chain, the relation of supply chain DEA to traditional ‘black box' DEA remains unclear. Moreover, the coordination effect between the two stages is ignored. In this paper, we define the coordination efficiency and propose a DEA-based approach for measuring the coordination effect of supply chain systems. We prove that the proposed multiplier DEA approach is equivalent to the envelopment DEA one. The proposed approach in this paper is verified via a numerical example from a supplier–manufacturer sustainable supply chain on resin companies finally.
Traditional data envelopment analysis (DEA) models mainly set the furthest targets as the frontier projection for inefficient decision-making units (DMUs). To achieve the efficient status with less effort, the closest target models are introduced which projected the least input/output improvement for inefficient DMUs. However, these works typically consider each DMU as a “black box” in the closest target setting, and thus these models cannot be directly extended to a system with network structure because there may be no efficient DMU for reference. This paper fills the gap by developing a closest target model for a two-stage system. Instead of constructing the efficient frontier only by the system efficient DMUs in the “black box” system, all the extreme efficient stages of the DMUs are considered to form the closest target for an inefficient DMU. Using our network closest target (NCT) model, a case of the 16 leading commercial banks in China is analyzed. The results show that these commercial banks performed steadily in both efficiency and input/output required improvement during the study period. Moreover, compared with the network furthest target (NFT) model, NCT requires each DMU less improvement on inputs/outputs, and usually obtain the higher efficiency. That is, NCT is more feasible, economical and optimistic. Lastly, we discuss the proportion of each dominating peer referred by the inefficient banks and the importance of these peers is ranked accordingly.
We develop a new network data envelopment analysis (DEA) approach for two-stage network systems considering a match between the production scale of the substages and the intermediate measure levels. Several explicit production axioms are introduced to build a production possibility set. New models are developed based on the production possibility set and a frontier projection procedure with the production scale matching process. Unlike the existing approach which assumes the intermediate measures are free-setting decision variables, the new envelopment network DEA models project the intermediate measures of a unit using the radial projection technique. Correspondingly, the resulting multiplier network DEA models allow for weight flexibility on the intermediate measures while holding a total value flow equivalence between a unit's two stages. We show that our approach does not suffer the known network DEA pitfalls. It identifies the overall efficiency, divisional efficiencies, and frontier projection using either an envelopment or a multiplier network DEA model, i.e., the primal-dual correspondence holds in our approach. Our approach also avoids uncertainties in determining divisional efficiencies by generating a unique pair of divisional efficiencies for each unit. Additionally, the adoption of the production scale matching process explains clearly the frontier projection procedure from the practical point of view. The proposed approach is illustrated with a numerical example and compared with the existing approaches with a case study of commercial bank branches.
Data envelopment analysis (DEA) is a non-parametric technique that is widely used in measuring the performance (efficiency) of decision-making units (DMUs). Many network DEA models have been built to investigate the internal structure of a system, which was considered a “black box” in traditional DEA models. However, the interdependent relationship between system components is rarely considered in performance evaluation. Interdependent processes system has become common in production systems because of complex competition. In this study, we build a novel DEA model to investigate the efficiency of a parallel system with two interdependent components. Furthermore, decentralized and centralized models are built to respectively measure the efficiency of DMUs in decentralized and centralized organization modes. The analysis shows that fewer inputs can produce more outputs but fewer wastes among the components under the centralized mode. Finally, our approach is verified through a numerical example and an application to Chinese high-level universities.
In a two-stage system with two divisions connected in series, fairly setting the target outputs for the first stage or equivalently the target inputs for the second stage is critical, in order to ensure that the two stages have incentives to collaborate with each other to achieve the best performance of the whole system. Data envelopment analysis (DEA) as a non-parametric approach for efficiency evaluation of multi-input, multi-output systems has drawn a lot of attention. Recently, many two-stage DEA models were developed for studying the internal structures of two-stage systems. However, there was no work studying the fair setting of the target intermediate products (or intermediate measures) although unreasonable setting will result in unfairness to the two stages because setting higher (fewer) intermediate measures means that the first (second) stage must make more efforts to achieve the overall production plan. In this chapter, a new DEA model taking account of fairness in the setting of the intermediate products is proposed, where the fairness is interpreted based on the Nash bargaining game model, in which the two stages negotiate their target efficiencies in the two-stage system based on their individual efficiencies. This approach is illustrated by an empirical application to insurance companies.
In the previous literature, it is demonstrated that the dual equivalence of multiplier and envelopment models that exists in standard data envelopment analysis (DEA) is not necessarily true for network DEA to derive frontier projection and divisional efficiency. Multiplier network model is often used for computing the divisional efficiency while envelopment network model is often used for identifying the frontier projection for inefficient decision making units (DMUs). In this paper, we show that the duality of standard DEA can be extended to two-stage additive network DEA. We propose an improved golden section method to solve parametric linear multiplier network model. Based on the primal-dual correspondence of parametric linear programming, we subsequently develop envelopment network model in parametric linear form to determine frontier projection and to find divisional efficiency as well.
This chapter first provides a summary of the existing results in the application of DEA on measuring efficiency of insurers in different countries in the world and explains how crosscountry comparisons can be conducted. The focus then narrows to the existing results in the analysis of efficiency of insurers in the Western Balkans countries. We then conduct the empirical research, using the 2014-2018 data for 16 insurance companies in Serbia, on the efficiency of life, non-life and mixed insurers, by means of the network DEA method, where in the first stage we measure the efficiency of insurers in accomplishing the risk pooling function and in the second stage we measure the efficiency of the investment process. Our results show that mixed insurers in Serbia are more efficient than specialized insurers, where non-life insurers are the least efficient. We also find that Serbian insurers perform better in terms of risk pooling efficiency, rather than in the investment efficiency. Technical efficiency in Serbia has declined in 2017 and 2018, which corresponds to deterioration of investment efficiency.
Dynamic data envelopment analysis (dynamic DEA) measures the relative efficiency of a set of decision-making units (DMUs) in multiple time periods. Efficiency improvement through time could be measured by learning curve concept, which provides a mathematical representation of the learning process when a task repetition occurs. It is insightful to measure the learning rates of DMUs individually as well as the whole efficient frontier. However, there is no method to provide the efficiency data required for calculating such learning rates in dynamic systems. In order to measure the learning rates of both efficient frontier and DMUs in a single DEA optimization run for each DMU, this paper proposes a new dynamic DEA model that calculates the efficiency of each period based on the frontier of every other period. To clarify the modeling approach, a multi-period non-dynamic model is presented before constructing the final dynamic model in primal and dual forms. The capabilities of the proposed model are then explored by a case study of 20 wells in the South Pars gas field, which is the first application of dynamic DEA in the oil and gas wells drilling performance analysis.
We show how to use DEA to model DMUs that produce in two stages, with output from the first stage becoming input to the second stage. Our model allows for any orientation or scale assumption. We apply the model to Major League Baseball, demonstrating its advantages over a standard DEA model. Our model detects inefficiencies that standard DEA models miss, and it can allow for resource consumption that the standard DEA model counts towards inefficiency. Additionally, our model distinguishes inefficiency in the first stage from that in the second stage, allowing managers to target inefficient stages of the production process.
This chapter focuses on health-care applications of DEA. The paper begins with a brief history of health applications and
discusses some of the models and the motivation behind the applications. Using DEA to develop quality frontiers in health
services is offered as a new and promising direction. The paper concludes with an eight-step application procedure and list
of do’s and don’ts when applying DEA to health services.
KeywordsHealth Services Research-Physicians-Hospitals-HMOs-Frontier Analysis-Health-Care Management-DEA-Performance-Efficiency-Quality
Network data envelopment analysis (DEA) concerns using the DEA technique to measure the relative efficiency of a system, taking into account its internal structure. The results are more meaningful and informative than those obtained from the conventional black-box approach, where the operations of the component processes are ignored. This paper reviews studies on network DEA by examining the models used and the structures of the network system of the problem being studied. This review highlights some directions for future studies from the methodological point of view, and is inspirational for exploring new areas of application from the empirical point of view.
A total of 180 consecutive testicular cancers containing a component of embryonal carcinoma (EC) were reviewed to assess the morphologic features of the EC component. EC mostly (84%) occurred as a component of a mixed germ cell tumor, but 16% were pure. Solid (55%), glandular (17%), and papillary (11%) were the most common primary patterns (predominant architectural pattern occupying at least 50%), whereas other less common primary patterns included nested (3%), micropapillary (2%), anastomosing glandular (1%), sieve-like glandular (<1%), pseudopapillary (<1%), and blastocyst-like (<1%). Occasionally, EC developed predominantly in the context of polyembryoma-like (6%) and diffuse embryoma-like ("necklace" pattern) (3%) proliferations. In all, 69% had secondary architectural patterns, the most frequent being glandular (31%), papillary (14%), and solid (12%). An appliqué appearance, in which smudged and degenerate-appearing EC cells appear "applied" to the tumor periphery, was common (67%). EC cells with clear cytoplasm and distinct cell membranes (seminoma-like) were present in 11%, and dense lymphocytic infiltration and granulomatous inflammation were seen in 7% and 3%, respectively. Features simulating yolk sac tumor and teratoma were also seen: pseudoendodermal sinuses (34%), columnar cells (20%), and secretory-type subnuclear cytoplasmic vacuoles (6%). Syncytiotrophoblast cells were frequent (46%). Intratubular EC, typically partly necrotic and calcified, occurred in 24%. The associated stroma was more often non-neoplastic (53%) than neoplastic (29%). The rarity of some poorly characterized patterns of EC (micropapillary, blastocyst-like, anastomosing glandular, and sieve-like glandular) and some that overlap with those of other germ cell tumors, as well as some uncommon cytologic features, may result in misinterpretation, potentially impacting management. The association with other more common patterns and typical cytologic features, together with simple awareness of these variant morphologies, are helpful in establishing an accurate diagnosis of EC.
Data envelopment analysis (DEA) is a method for measuring the efficiency of peer decision making units (DMUs). Recently network DEA models been developed to examine the efficiency of DMUs with internal structures. The internal network structures range from a simple two-stage process to a complex system where multiple divisions are linked together with intermediate measures. In general, there are two types of network DEA models. One is developed under the standard multiplier DEA models based upon the DEA ratio efficiency, and the other under the envelopment DEA models based upon production possibility sets. While the multiplier and envelopment DEA models are dual models and equivalent under the standard DEA, such is not necessarily true for the two types of network DEA models. Pitfalls in network DEA are discussed with respect to the determination of divisional efficiency, frontier type, and projections. We point out that the envelopment-based network DEA model should be used for determining the frontier projection for inefficient DMUs while the multiplier-based network DEA model should be used for determining the divisional efficiency. Finally, we demonstrate that under general network structures, the multiplier and envelopment network DEA models are two different approaches. The divisional efficiency obtained from the multiplier network DEA model can be infeasible in the envelopment network DEA model. This indicates that these two types of network DEA models use different concepts of efficiency. We further demonstrate that the envelopment model’s divisional efficiency may actually be the overall efficiency.
Data envelopment analysis (DEA) is a method for measuring the efficiency of peer decision making units (DMUs). This tool has been utilized by a number of authors to examine two-stage processes, where all the outputs from the first stage are the only inputs to the second stage. The current article examines and extends these models using game theory concepts. The resulting models are linear, and imply an efficiency decomposition where the overall efficiency of the two-stage process is a product of the efficiencies of the two individual stages. When there is only one intermediate measure connecting the two stages, both the noncooperative and centralized models yield the same results as applying the standard DEA model to the two stages separately. As a result, the efficiency decomposition is unique. While the noncooperative approach yields a unique efficiency decomposition under multiple intermediate measures, the centralized approach is likely to yield multiple decompositions. Models are developed to test whether the efficiency decomposition arising from the centralized approach is unique. The relations among the noncooperative, centralized, and standard DEA approaches are investigated. Two real world data sets and a randomly generated data set are used to demonstrate the models and verify our findings. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008
In management contexts, mathematical programming is usually used to evaluate a collection of possible alternative courses of action en route to selecting one which is best. In this capacity, mathematical programming serves as a planning aid to management. Data Envelopment Analysis reverses this role and employs mathematical programming to obtain ex post facto evaluations of the relative efficiency of management accomplishments, however they may have been planned or executed. Mathematical programming is thereby extended for use as a tool for control and evaluation of past accomplishments as well as a tool to aid in planning future activities. The CCR ratio form introduced by Charnes, Cooper and Rhodes, as part of their Data Envelopment Analysis approach, comprehends both technical and scale inefficiencies via the optimal value of the ratio form, as obtained directly from the data without requiring a priori specification of weights and/or explicit delineation of assumed functional forms of relations between inputs and outputs. A separation into technical and scale efficiencies is accomplished by the methods developed in this paper without altering the latter conditions for use of DEA directly on observational data. Technical inefficiencies are identified with failures to achieve best possible output levels and/or usage of excessive amounts of inputs. Methods for identifying and correcting the magnitudes of these inefficiencies, as supplied in prior work, are illustrated. In the present paper, a new separate variable is introduced which makes it possible to determine whether operations were conducted in regions of increasing, constant or decreasing returns to scale (in multiple input and multiple output situations). The results are discussed and related not only to classical (single output) economics but also to more modern versions of economics which are identified with "contestable market theories."
Utilizing recent developments in data envelopment analysis (DEA), this paper examines the performance of the top 55 U.S. commercial banks via a two-stage production process that separates profitability and marketability. Substantial performance inefficiency is uncovered in both dimensions. Relatively large banks exhibit better performance on profitability, whereas smaller banks tend to perform better with respect to marketability. New context-dependent performance measures are defined for profitability and marketability which employ a DEA stratification model and a DEA attractiveness measure. When combined with the original DEA measure, the context-dependent performance measure better characterizes the profitability and marketability of 55 U.S. commercial banks. The new approach identifies areas for improved bank performance over the two-stage production process. The effect of acquisition on efficiency and attractiveness is also examined.
The efficiency of decision processes which can be divided into two stages has been measured for the whole process as well as for each stage independently by using the conventional data envelopment analysis (DEA) methodology in order to identify the causes of inefficiency. This paper modifies the conventional DEA model by taking into account the series relationship of the two sub-processes within the whole process. Under this framework, the efficiency of the whole process can be decomposed into the product of the efficiencies of the two sub-processes. In addition to this sound mathematical property, the case of Taiwanese non-life insurance companies shows that some unusual results which have appeared in the independent model do not exist in the relational model. In other words, the relational model developed in this paper is more reliable in measuring the efficiencies and consequently is capable of identifying the causes of inefficiency more accurately. Based on the structure of the model, the idea of efficiency decomposition can be extended to systems composed of multiple stages connected in series.
A nonlinear (nonconvex) programming model provides a new definition of efficiency for use in evaluating activities of not-for-profit entities participating in public programs. A scalar measure of the efficiency of each participating unit is thereby provided, along with methods for objectively determining weights by reference to the observational data for the multiple outputs and multiple inputs that characterize such programs. Equivalences are established to ordinary linear programming models for effecting computations. The duals to these linear programming models provide a new way for estimating extremal relations from observational data. Connections between engineering and economic approaches to efficiency are delineated along with new interpretations and ways of using them in evaluating and controlling managerial behavior in public programs.
We classify the contributions of DEA literature assessing Decision Making Units (DMUs) whose internal structure is known.
Starting from an elementary framework, we define the main research areas as shared flow, multilevel and network models, depending
on the assumptions they are subject to. For each model category, the principal mathematical formulations are introduced along
with their main variants, extensions and applications. We also discuss the results of aggregating efficiency measures and
of considering DMUs as submitted to a central authority that imposes constraints or targets on them. A common feature among
the several models is that the efficiency evaluation of the DMU depends on the efficiency values of its subunits thereby increasing
the discrimination power of DEA methodology with respect to the black box approach.
Data envelopment analysis (DEA) is a method for measuring the efficiency of peer decision making units (DMUs). An important area of development in recent years has been devoted to applications wherein DMUs represent two-stage or network processes. One particular subset of such processes is those in which all the outputs from the first stage are the only inputs to the second stage. The current paper reviews these models and establishes relations among various approaches. We show that all the existing approaches can be categorized as using either Stackelberg (leader-follower), or cooperative game concepts. Future perspectives and challenges are discussed.
Data envelopment analysis (DEA) is a method for measuring the efficiency of peer decision making units (DMUs). Recently DEA has been extended to examine the efficiency of two-stage processes, where all the outputs from the first stage are intermediate measures that make up the inputs to the second stage. The resulting two-stage DEA model provides not only an overall efficiency score for the entire process, but as well yields an efficiency score for each of the individual stages. Due to the existence of intermediate measures, the usual procedure of adjusting the inputs or outputs by the efficiency scores, as in the standard DEA approach, does not necessarily yield a frontier projection. The current paper develops an approach for determining the frontier points for inefficient DMUs within the framework of two-stage DEA.