Directional Distance Function Model Research Articles

On indication, strict monotonicity, and efficiency of projections in a general class of path-based data envelopment analysis models

Data envelopment analysis (DEA) theory formulates a number of desirable properties that DEA models should satisfy. Among these, indication, strict monotonicity, and strong efficiency of projections tend to be grouped together in the sense that, in individual models, typically, either all three are satisfied or all three fail at the same time. Specifically, in slacks-based graph models, the three properties are always met; in path-based models, such as radial models, directional distance function models, and the hyperbolic function model, the three properties, with some minor exceptions, typically all fail.Motivated by this observation, the article examines relationships among indication, strict monotonicity, and strong efficiency of projections in the class of path-based models over variable returns-to-scale technology sets. Under mild assumptions, it is shown that the property of strict monotonicity and strong efficiency of projections are equivalent, and that both properties imply indication. This paper also characterises a narrow class of technology sets and path directions for which the three properties hold in path-based models.

Collaborative management of environmental pollution and carbon emissions drives local green growth: An analysis based on spatial effects

Collaborative management of environmental pollution and carbon emissions (CMPC) has been a major policy instrument to promote Sustainable Development Goals (SDG) in recent years. However, the relationship between the benefits and drawbacks of this environmental management practice for green growth in and around a local area remains to be clarified. Using 30 provinces in China during 2001–2019 as the object of analysis, we assessed the efficiency of local CMPC practices using the nonradial directional distance function (NDDF) model, predicted local green growth using the frontier green complexity index (GCI), and empirically examined the spatial effects, locational heterogeneity, and threshold characteristics of the relationship using the spatial Durbin model and the panel threshold model. Our study finds that although efficient CMPC does drive local green growth, the promotion effect is nonlinear with decreasing marginal effect. This effect is particularly obvious in economically developed regions with higher CMPCs, which will absorb resources from neighboring regions and create a "siphoning" effect. It was found that local financial support and foreign direct investment (FDI) can radiate green growth to neighboring regions; therefore, CMPC practice needs to pay more attention to the effect of joint governance, supplemented by financial and foreign investment policy tools, to better promote the green transformation of local economy.

A double bootstrap data envelopment analysis model with stochastic reducible outputs and expandable inputs: An application to power plants

Clean production of electricity is not only cost-effective but also effective in reducing pollutants. Toward this end, the use of clean fuels is strongly recommended by environmentalists. Benchmarking techniques, especially data envelopment analysis, are an appropriate tool for measuring the relative efficiency of firms with environmental pollutants. In classic data envelopment analysis models, decision-makers are faced with production processes in which reducible inputs are used to produce expandable outputs. In this contribution, we consider production processes when the input and output data are given in stochastic form and some throughputs are reducible and some others are expandable. A stochastic directional distance function model is proposed to calculate the relative technical efficiency of firms. In order to evaluate firm-specific technical efficiency, we apply bootstrap DEA. We first calculate the technical efficiency scores of firms by classic DEA model. Then, the double bootstrap DEA model is applied to determine the impact of explanatory variables on firm efficiency. To demonstrate the applicability of the procedure, we present an empirical application wherein we employ power plants.

The impact of repayment obligations arising as a by‐product of input use on partial inefficiency: Evidence from Western Australian farm businesses

AbstractFarm businesses often use debt to finance the purchase inputs and meet operational costs. Selected studies have investigated the impact of debt use on farm performance as measured using technical efficiency. However, no prior studies in agriculture have considered treating the debt repayment obligation created as a by‐product of production when benchmarking farm performance. This study employs nonparametric directional distance function models to quantify the impact of debt repayment as a by‐product of input use purchased on credit on farm‐level partial inefficiency using a panel data of 54 mixed enterprise broadacre farms in Western Australia from 2002 to 2011. The study finds that omitting repayment obligations created in the production process in analysis results in underestimating partial inefficiency scores. Farm size, production diversification and rainfall zone are significant drivers of partial inefficiency. The implication of the study's findings is that failure to account for repayment obligations created in the production process may result in erroneous policy suggestions that undermine the efforts of farm businesses to minimise or avoid financial stress.

On modelling non‐performing loans in bank efficiency analysis

AbstractThis paper introduces a methodological framework for the examination of non‐performing loans (NPLs) as reverse outputs under the extended strong disposability assumption, which does not require NPLs to be jointly produced with net loans, as it is implied when they are modelled as undesirable outputs. A directional distance function model with reverse outputs is used and is compared with the models that treat NPLs as an undesirable output under the weak disposability and the constrained weak disposability assumptions with uniform and non‐uniform abatement factors. The model is applied at the case of European banks and for the sample to be representative the banks are chosen based on the European Banking Authority (EBA) stress test of 2021. The results indicate that the reverse output model have greater discriminatory power relative to all other models.

Natural resource and energy efficiency in BRICS countries: Can green innovation capabilities really change the game?

The inefficient exploitation of resources, overdependence on fossil fuels for energy production, and inappropriate waste management strategies have resulted in mounting environmental challenges that seriously threaten humanity and the planet.The United Nations urges businesses to transition from fossil fuels to sustainable practices, fostering cooperation between natural resources, the economy, and ecology to achieve SDGs. This shift is critical for emerging economies, which may experience rapid growth while heavily relying on fossil fuels. This study uses the nonparametric data envelopment analysis and directional distance function (DEA-DFF) model to examine how organizational green and technological innovation capabilities impact natural resource and renewable energy efficiency in BRICS countries from 2011 to 2021. The results indicated that green and technological innovation significantly improve resource and energy efficiency in BRICS economies, supporting the theoretical arguments given in the dynamic capabilities theory (DCT), business technology adoption model (BTAM), and natural resource-based view (NRBV). It all proposes theoretical and practical implications for similar developing nations. Overall, the results of the study are robust to different econometric models and data processing techniques.

The impact of water rights trading on water resource use efficiency: Evidence from China's water rights trading pilots

With the increasing conflict between the supply and demand of water resources, enhancing water resource use efficiency has gradually become a focus of attention. As a new way to manage water resources, water rights trading pilots have been implemented in China. It is important to explore the impact of water rights trading on regional water resource use efficiency and possible impact paths. This paper employs a global nonradial directional distance function (GNDDF) model to measure water resource use efficiency, which is more accurate and reliable. To discern the effect of China's water rights trading pilot policy on enhancing regional water resource use efficiency, we apply the regression control method (RCM) for counterfactual analysis and test the possible pathways through which the water rights trading policy affects water resource use efficiency. This article draws the following conclusions. (i) China's overall water resource use efficiency has not reached a high level over the past decade, and water resource use efficiency has exhibited substantial regional disparities. (ii) There is some evidence to suggest that the water rights trading policy may have a positive impact on improving water resource use efficiency in the pilot provinces. However, the improvement effect exhibits regional variations, which are likely attributed to the initial water resource use levels in the pilot provinces and the specific implementation modes of water rights trading. (iii) Water rights trading policies can enhance water resource use efficiency through two pathways, including facilitating reclaimed water reuse and promoting water use restructuring.

A multi-attribute group decision-making method based on data envelopment analysis under parallel expert evaluation system

In traditional multi-attribute group decision-making (MAGDM) methods, the aggregation of expert evaluation information, the normalization of attribute scales, and the determination of attribute prior weights can lead to information loss and inconsistent results. In addition, these methods cannot identify and improve the shortcomings of non-optimal alternatives. Considering these, this paper proposes a MAGDM method based on data envelopment analysis (DEA) by setting up a parallel expert evaluation system, thus integrating the initial opinions of all experts to rank the alternatives from a global perspective. First, we construct the multi-expert evaluation process as a parallel system from the global perspective so that the decision-making process is based on the initial expert evaluation information. Then, a new directional distance function model based on endogenous projection directions is proposed to measure the alternatives under each expert subsystem, eliminating data processing and artificial weight assignment. The non-optimal alternatives can be improved to an efficient state by adjusting the attribute levels to target values. Next, we propose a DDF super-efficiency model to perform a full ranking of the alternatives. Finally, a case study is provided to demonstrate the rationality and effectiveness of the proposed method.

Carbon emission efficiency of 284 cities in China based on machine learning approach: Driving factors and regional heterogeneity

The rational categorization and assessment of carbon emission efficiency (CEE) and its drivers are crucial for coping with the global climate crisis. To address the bias of univariate modeling and challenge of ignoring the heterogeneity of drivers across cities, this study explores differences between carbon emission drivers across different types of cities and regions to reveal the spatial distribution characteristics of urban CEE and heterogeneity of emission reduction potential. We use a non-radial, non-directional relaxation measure-based directional distance function (SBM-DDF) model to assess the CEE of 284 cities over the period from 2006 to 2020. Machine-learning algorithms are applied to identify city characteristics to determine the effects of city- development types and their characteristic drivers. The results of the driver analysis show that energy consumption, gross regional product, spatial area, and population size are the key factors influencing in the heterogeneity of cities' CEE, with an importance ranking of 0.578, 0.507, 0.432, and 0.418, respectively. The results of for the heterogeneity of the cities' heterogeneity further confirm that energy consumption has the greatest impact on energy-dependent cities (EDCs), economic-development cities (ECDCs), and low-carbon potential cities (LPCs), whereas among the Low-carbon growth cities (LCGs), science, technology, and innovation, urban greening, and electricity consumption play an important roles in promoting greening and low- carbon development, which can help to determine the low- carbon development model for each type of city. Finally, energy consumption affects cities in the central region more than in the eastern and western regions. Based on the results of estimating the heterogeneity of urban carbon- emission rates, we propose customized emission- reduction development pathways to guide urban low-carbon development and formulate carbon- reduction policies.

Research on Dynamic Evolutionary Efficiency and Regional Differentiation of High-Tech Industrial Chain Networks

This research dynamically evaluates the innovation efficiency of China’s high-tech industry and explores the efficiency differences in basic innovation, application innovation, and income innovation of this industry for different regions. Based on panel data of 30 provinces from 2015 to 2019, we construct a three-stage dynamic DDF (Directional Distance Function) model, divide the high-tech industry into three stages and measure the efficiency of the three stages. From 2015 to 2019, most of the total efficiency of China’s high-tech industries showed an upward trend, with the western region having the highest total efficiency value and the central region the lowest. Most of the 30 provinces present basic innovation efficiency < applied innovation efficiency < profitable innovation efficiency. The conclusions from the article’s empirical analysis can help developing countries concerned find out key links to improve the efficiency of high-tech entrepreneurship and innovation and then formulate relevant industrial policies.

Local government competition, new energy industry agglomeration and urban ecological total factor energy efficiency: A new perspective from the role of knowledge

The contradiction between industrial agglomeration and energy constrains the development of cities. Government competition can affect the flow of factors for industrial agglomeration. Based on the data of 274 cities in China from 2005 to 2020, this paper constructs improved non-radial directional distance function (NDDF) and dynamic spatial durbin model (SDM). The paper further uses a threshold effect model and a mediating effects model to examines the impact and spillover effects of local government competition (LGC), new energy industry clustering and their integration on ecological total factor energy efficiency (ETFEE) from the perspective of the role of knowledge. To solve the problem of endogeneity, this paper uses the presence or absence of highway entrances and exits in cities in 2004 as an instrumental variable. The study found that new energy industry agglomeration and LGC can promote ETFEE in the short term, but it will inhibit ETFEE in the long term. New energy industry agglomeration affects local ETFEE through two types of mediating effects: narrowing knowledge breadth and widening knowledge distance between cities. At the same time, new energy industry agglomeration and LGC has a significant spatial spillover effect on urban ETFEE in the short term. Areas with high urban ETFEE can promote the improvement of ETFEE in neighbouring areas. However, from the perspective of regional differences, it finds that the concentration of new energy industries and LGC have different effects on ETFEE. There is a significant impact in the eastern region, but not in the central and western regions. Finally, the paper gives constructive advice for the adaptation of the knowledge structure and the improvement of energy efficiency through the national and governmental levels.

Agricultural tridimension pollution emission efficiency in China: An evaluation system and influencing factors

We innovated traditional one-sided agricultural carbon emission efficiency research, comprehensively explored the agricultural tridimension pollution emission efficiency (ATPEE), constructed an ATPEE evaluation system considering technological heterogeneity characteristics based on the meta-frontier nonradial directional distance function (NDDF) model in a total-factor framework, evaluated the ATPEE in 30 mainland China provinces from 1997 to 2021 for the first time, and empirically studied the factors influencing the ATPEE in China with the Tobit model. The results showed the following: (1) ATPEE improvement potentials of 75.16 % and 50.88 % occur under the meta and group frontiers, respectively. (2) The eastern region represents the potential optimal agricultural tridimension pollution emission technology, while the central and western regions exhibit a large gap with the national potential optimal technology level. (3) The ATPEE loss in the eastern and western regions mainly results from management inefficiency, while that in the central region largely results from technology gap inefficiency. (4) The effects of the industrialization rate, urban–rural income gap, agricultural production structure, financial support for agriculture, natural conditions and effective irrigation rate on the ATPEE in the different regions vary.

Balancing crop security and sustainable cropland use: Policy lessons from the Watershed Ecosystem Service Payments in Xin’an River, China

This study investigates the impact of Payments for Watershed Ecosystem Services (PWS) on the sustainable utilisation of cropland and crop security in Huangshan City, which is the primary implementation area for the Payments for Xin’an River Watershed Ecosystem Services pilot in China. Using a two-stage analytical framework along with a quasi-natural experiment, the research examines the policy’s effectiveness in Huangshan City. In the initial stage, the directional distance function and global Malmquist-Luenberger index (DDF-GML) model are employed to calculate the agricultural green total factor productivity (AGTFP), representing sustainable management of cultivated land. Concurrently, the study constructs unobservable counterfactual paths using the synthetic control method (SCM) to evaluate impacts on land use practices. The findings show an increase in Huangshan City’s AGTFP from 1.118 in 2010 to 1.823 in 2020. Quantitative assessments indicate that the policy contributed to a 15.3% rise in AGTFP in the PWS cohort compared to the control group. Though initial decreases were detected during three pilot intervals, a consistent upward trajectory in AGTFP was identified, suggesting that PWS encourages crop stability and the sustainable development of arable land. Further support from permutation, regional, and temporal placebo examinations affirm these conclusions. This work implies that PWS may be extended to other agriculturally vulnerable regions to foster a balanced alignment between sustainable growth and the enhancement of crop security The assessment focuses on policy lessons that could inform broader sustainability strategies.

Renewable energy performance analysis using fuzzy dynamic directional distance function model under natural and managerial disposability

Addressing the dynamic performance aspects of processes and identifying the sources of inefficiency are key considerations for making informed decisions, especially in situations where imprecise data are available. This study aims to evaluate the performance of renewable energy using the Fuzzy Dynamic Directional Distance Function (FDDDF) model considering natural and managerial disposability for 34 OECD member countries. As a non-parametric methodology, Data Envelopment Analysis (DEA) is used to measure dynamic fuzzy environmental efficiencies of renewable energy through multiple fuzzy inputs and outputs. Using the proposed approach, optimistic, pessimistic and integrated renewable energy performances of entities with carry-overs are assessed for each period and generally. The empirical findings reveal that the proposed FDDDF approach has strong capability to discriminate the performance of renewable energy.

What Is the State of Development of Eco-Wellbeing Performance in China? An Analysis from a Three-Stage Network Perspective

After rapid development in recent decades, China has laid a strong economic foundation and established material conditions. Despite this, the country still confronts a range of challenges that impede higher-quality economic and social development. To measure high-quality regional development, eco-wellbeing performance (EWP) has become an important tool that seeks to strike a balance between economic development, human wellbeing, and environmental protection. This paper proposes a three-stage network efficiency evaluation model to decompose EWP into three stages, namely ecological economic efficiency (EEE), economic innovation efficiency (EIE), and innovation wellbeing efficiency (IWE). A directional distance function (DDF) model was utilized to assess the overall EWP efficiency and phase-in efficiency for 248 cities over the period from 2016 to 2020. The study results indicate that EWP in China is generally low. In terms of the three sub-stages of EWP, the first stage performs optimally, the third stage performs the second best, and the second stage is the worst. The research suggests that the principal reason for the generally low EWP may be linked to the efficiency of the transformation of economic investments into innovative outcomes in the second stage. From a regional viewpoint, EWP generally shows a gradually decreasing trend from the east, central, and west to the northeast, but the stage performance varies among regions. The eastern region has lower EEE, but both EIE and IWE perform better than the national average; the central region is well-balanced between the three stages; the western region leads the country in EEE, but lags in both EIE and IWE; and the northeast region has inferior performance in all stages. This study’s findings provide vital reference values for policy-makers to determine key points for enhancing EWP in different regions of China.

Carbon abatement in the European chemical industry: assessing the feasibility of abatement technologies by estimating firm-level marginal abatement costs

As many other industries, the chemical industry has to strongly reduce its' carbon emissions, for which various abatement technologies exist. We study the economic feasibility of these abatement technologies by estimating the Marginal Abatement Cost (MAC) of CO2 emissions for 24 firms in the European chemical sector over the period 2015–2020. We estimate the firm-level MAC by using a quadratic directional output distance function (DDF) model. We find a median MAC of 429 €/t CO2, which is significantly above current carbon prices, indicating that most firms in this industry prefer to use carbon allowances instead of reducing their own emissions. We conclude that carbon abatement in the chemical industry is only likely when the carbon price is significantly higher or when financial support is provided for certain abatement technologies, such as renewable hydrogen and bio-based ammonia.

Measuring efficiency of safe work environment from the perspective of the decent work Agenda

Decent Work Agenda consists of a comprehensive initiative for promoting safety at work and social protection. Over 20 years since its conceptual release, measuring the progress of its elements is still challenging even after the publication of the decent work indicators guideline by the International Labour Organization in 2012. To close this gap, we use a Directional Distance Function (DDF) to measure the efficiency of safe work environment, and propose a combined measure taking into consideration also the efficacy. To illustrate the application of DDF in a reality-based case, we conducted a longitudinal study in a multinational organization. Data were collected from 21 branches of the company over 4 years (2018–2021). In the period of analysis, 60% of the branches were efficient in average, composing an overall efficiency score of 0.91. Our results also indicated low dispersion between the efficiency scores year on year due to a small-scale interquartile range. Finally, the use of efficiency combined with efficacy resulted in a promising approach for managerial applications. This research presents some contributions. One is the novelty approach of measuring the efficiency of safe work environment using a DDF model in a real-world application. Another is the managerial benefits of identifying benchmarks, as well as revealing potential improvements as a mechanism to reduce decent work deficits. From a modeling perspective, our conclusions suggest caution in considering only efficiency to measure safe work environment due to its relative nature. Thus, further studies are recommended to explore the use of combined measures in the analysis of decent work.

Impact of China’s Low-Carbon City Pilot Policies on Enterprise Energy Efficiency

An increase in energy efficiency is an essential element and a powerful driving force for the in-depth implementation of the sustainable development strategies necessary in accelerating the promotion of green, circular, and low-carbon development, as well as to promote the comprehensive green transformation of economic and social development. An important question with regard to this paper is thus: can the low-carbon city pilot policy promote energy efficiency improvement, and if so, through what mechanisms? This paper uses the SBM–Malmquist–Luenberger index method to measure the green total factor energy efficiency and examines the impact and pathways of the pilot policy on the energy efficiency of enterprises, using a sample of listed manufacturing enterprises in 230 prefecture-level cities in China from 2007 to 2020. Additionally, the time-varying difference-in-differences (DID) method is approached in this paper. After replacing energy efficiency with slack-based measure directional distance function model (SBM-DDF) and conducting a series of robustness tests, this study found that the pilot policy can significantly improve the energy efficiency of manufacturing enterprises. A mechanism test shows that this policy can promote green innovation effect and agglomeration effect to improve enterprises’ energy efficiency. The low-carbon city pilot policy has contributed the most to energy efficiency through enterprise investment in green innovation and manufacturing agglomeration. Heterogeneity analysis found that policy effect differs among firms in terms of different sizes and properties, and the pilot policy plays different roles among different regions. This paper provides firm-level theoretical support and empirical evidence for evaluating low-carbon city pilot policy and offers policy recommendations.

A unified approach to radial, hyperbolic, and directional efficiency measurement in data envelopment analysis

The paper analyses properties of a large class of “path-based” Data Envelopment Analysis models through a unifying general scheme. The scheme includes the well-known oriented radial models, the hyperbolic distance function model, the directional distance function models, and even permits their generalisations. The modelling is not constrained to non-negative data and is flexible enough to accommodate variants of standard models over arbitrary data.Mathematical tools developed in the paper allow systematic analysis of the models from the point of view of ten desirable properties. It is shown that some of the properties are satisfied (resp., fail) for all models in the general scheme, while others have a more nuanced behaviour and must be assessed individually in each model. Our results can help researchers and practitioners navigate among the different models and apply the models to mixed data.

A generalized robust data envelopment analysis model based on directional distance function

In the literature of data envelopment analysis, the directional distance function (DDF) model is commonly used to measure efficiency improvement, as it allows the decision-maker to choose an appropriate direction that permits input contraction and output expansion. However, choosing the right direction is challenging in empirical research. Additionally, efficiency measurement becomes problematic when input and output data are uncertain. To address these issues, we present an equivalent DDF model in multiplier form and use the robust optimization approach to construct a technology in order to develop a generalized robust-DDF measure of efficiency. Among the three commonly used predefined directions (input-oriented, output-oriented, and proportional) considered in this study, we define the robust direction as the one with the minimum price that decision-maker must pay to be immune to data uncertainty. To demonstrate the usefulness of our proposed robust direction measure, we apply it a real-life data on life insurance companies in India over eight years (2011–12–2018–19). Our results show that the proportional direction exhibits the lowest price of robustness and is therefore the most appropriate for measuring potential efficiency improvement. Additionally, the increasing efficiency trend in the life insurance industry confirms the evidence of increased work intensity due to competition resulting from insurance reforms, supporting the competition and X-efficiency hypothesis.